ALGEBRAIC GRAPH THEORY - cloudflare-ipfs.com

ALGEBRAIC GRAPH THEORY


Second Edition

NORMAN BIGGS

London School of Economics

CAMBRIDGEUNIVERSITY PRESS

CAMBRIDGE u n i v e r s i t y p r e s s

Cambridge New York Melbourne Madrid Cape Town SingaporeSatildeo Paulo Delhi Dubai Tokyo Mexico City

Cambridge University PressThe Edinburgh Building Cambridge CB2 8RU UK

Published in the United States of America byCambridge University Press New York

wwwcambridgeorgInformation on this title wwwcambridgeorg9780521458979

copy Cambridge University Press 1974 1993

This publication is in copyright Subject to statutory exceptionand to the provisions of relevant collective licensing agreementsno reproduction of any part may take place without the writtenpermission of Cambridge University Press

First published 1974Second edition 1993Reprinted 1996

A catalogue record for this publication is available from the British Library

Library of Congress cataloguing in publication data available

ISBN 978-0-521-20335-7 Hardback

ISBN 978-0-521-45897-9 Paperback

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred to inthis publication and does not guarantee that any content on such websites isor will remain accurate or appropriate Information regarding prices traveltimetables and other factual information given in this work is correct atthe time of first printing but Cambridge University Press does not guaranteethe accuracy of such information thereafter

Contents

Preface vii

1 Introduction 1

PART ONE - LINEAR ALGEBRA IN GRAPH THEORY

2 The spectrum of a graph 73 Regular graphs and line graphs 144 Cycles and cuts 235 Spanning trees and associated structures 316 The tree-number 387 Deteminant expansions 448 Vertex-partitions and the spectrum 52

PART TWO - COLOURING PROBLEMS

9 The chromatic polynomial 6310 Subgraph expansions 7311 The multiplicative expansion 8112 The induced subgraph expansion 8913 The Tutte polynomial 9714 Chromatic polynomials and spanning trees 106

PART THREE - SYMMETRY AND REGULARITY

15 Automorphisms of graphs 11516 Vertex-transitive graphs 12217 Symmetric graphs 130

vi Contents

18 Symmetric graphs of degree three 13819 The covering-graph construction 14920 Distance-transitive graphs 15521 Feasibility of intersection arrays 16422 Imprimitivity 17323 Minimal regular graphs with given girth 180

References 191Index 202

Preface

This book is a substantially enlarged version of the Cambridge Tractwith the same title published in 1974 There are two major changes

bull The main text has been thoroughly revised in order to clarify theexposition and to bring the notation into line with current practiceIn the course of revision it was a pleasant surprise to find that theoriginal text remained a fairly good introduction to the subject bothin outline and in detail For this reason I have resisted the temptationto reorganise the material in order to make the book rather more like astandard textbook

bull Many Additional Results are now included at the end of eachchapter These replace the rather patchy selection in the old versionand they are intended to cover most of the major advances in the lasttwenty years It is hoped that the combination of the revised text andthe additional results will render the book of service to a wide range ofreaders

I am grateful to all those people who have helped by commenting uponthe old version and the draft of the new one Particular thanks are dueto Peter Rowlinson Tony Gardiner Ian Anderson Robin Wilson andGraham Brightwell On the practical side I thank Alison Adcock whoprepared a TgX version of the old book and David Tranah of CambridgeUniversity Press who has been constant in his support

Norman Biggs March 1993

Introduction to algebraic graph theory

About the book

This book is concerned with the use of algebraic techniques in the studyof graphs The aim is to translate properties of graphs into algebraicproperties and then using the results and methods of algebra to deducetheorems about graphs

It is fortunate that the basic terminology of graph theory has now be-come part of the vocabulary of most people who have a serious interestin studying mathematics at this level A few basic definitions are gath-ered together at the end of this chapter for the sake of convenience andstandardization Brief explanations of other graph-theoretical terms areincluded as they are needed A small number of concepts from matrixtheory permutation-group theory and other areas of mathematics areused and these are also accompanied by a brief explanation

The literature of algebraic graph theory itself has grown enormouslysince 1974 when the original version of this book was published Liter-ally thousands of research papers have appeared and the most relevantones are cited here both in the main text and in the Additional Re-sults at the end of each chapter But no attempt has been made toprovide a complete bibliography partly because there are now severalbooks dealing with aspects of this subject In particular there are twobooks which contain massive quantities of information and on which itis convenient to rely for amplification and exemplification of the mainresults discussed here

2 Introduction to algebraic graph theory

These are

Spectra of Graphs DM Cvetkovic M Doob and H Sachs AcademicPress (New York) 1980Distance-Regular Graphs AE Brouwer AM Cohen and A NeumaierSpringer-Verlag (Berlin) 1989References to these two books are given in the form [CvDS p 777] and[BCN p 888]

CD Godsils recent book Algebraic Combinatorics (Chapman andHall 1993) arrived too late to be quoted as reference It is in manyways complementary to this book since it covers several of the sametopics from a different point of view Finally the long-awaited Handbookof Combinatorics will contain authoritative accounts of many subjectsdiscussed in these pages

Outline of the book

The book is in three parts each divided into a number of short chap-ters The first part deals with the applications of linear algebra andmatrix theory to the study of graphs We begin by introducing the ad-jacency matrix of a graph this matrix completely determines the graphand its spectral properties are shown to be related to properties of thegraph For example if a graph is regular then the eigenvalues of itsadjacency matrix are bounded in absolute value by the degree of thegraph In the case of a line graph there is a strong lower bound for theeigenvalues Another matrix which completely describes a graph is theincidence matrix of the graph This matrix represents a linear mappingwhich determines the homology of the graph The problem of choosinga basis for the homology of a graph is just that of finding a fundamentalsystem of cycles and this problem is solved by using a spanning treeAt the same time we study cuts in the graph These ideas are thenapplied to the systematic solution of network equations a topic whichsupplied the stimulus for the original theoretical development We theninvestigate formulae for the number of spanning trees in a graph andresults which are derived from the expansion of determinants Theseexpansions illuminate the relationship between a graph and the charac-teristic polynomial of its adjacency matrix The first part ends with adiscussion of how spectral techniques can be used in problems involvingpartitions of the vertex-set such as the vertex-colouring problem

The second part of the book deals with the colouring problem from adifferent point of view The algebraic technique for counting the colour-ings of a graph is founded on a polynomial known as the chromatic

Introduction to algebraic graph theory 3

polynomial We first discuss some simple ways of calculating this poly-nomial and show how these can be applied in several important casesMany important properties of the chromatic polynomial of a graph stemfrom its connection with the family of subgraphs of the graph and weshow how the chromatic polynomial can be expanded in terms of sub-graphs From the first (additive) expansion another (multiplicative)expansion can be derived and the latter depends upon a very restrictedclass of subgraphs This leads to efficient methods for approximatingthe chromatic polynomials of large graphs A completely different kindof expansion relates the chromatic polynomial to the spanning trees of agraph this expansion has several remarkable features and leads to newways of looking at the colouring problems and some new properties ofchromatic polynomials

The third part of the book is concerned with symmetry and regularityproperties A symmetry property of a graph is related to the existenceof automorphisms - that is permutations of the vertices which pre-serve adjacency A regularity property is defined in purely numericalterms Consequently symmetry properties induce regularity propertiesbut the converse is not necessarily true We first study the elementaryproperties of automorphisms and explain the connection between theautomorphisms of a graph and the eigenvalues of its adjacency matrixWe then introduce a hierarchy of symmetry conditions which can beimposed on a graph and proceed to investigate their consequences Thecondition that all vertices be alike (under the action of the group of auto-morphisms) turns out to be rather a weak one but a slight strengtheningof it leads to highly non-trivial conclusions In fact under certain condi-tions there is an absolute bound to the level of symmetry which a graphcan possess A strong symmetry property called distance-transitivityand the consequent regularity property called distance-regularity arethen introduced We return to the methods of linear algebra to derivenumerical constraints upon the existence of graphs with these propertiesFinally these constraints are applied to the problem of finding minimalregular graphs whose degree and girth are given

Basic definitions and notation

Formally a general graph F consists of three things a set VT a set poundTand an incidence relation that is a subset of VT x ET An elementof VT is called a vertex an element of poundT is called an edge and theincidence relation is required to be such that an edge is incident witheither one vertex (in which case it is a loop) or two vertices If every


edge is incident with two vertices and no two edges are incident withthe same pair of vertices then we say that F is a strict graph or brieflya graph In this case ET can be regarded as a subset of the set ofunordered pairs of vertices We shall deal mainly with graphs (that isstrict graphs) except in Part Two where it is sometimes essential toconsider general graphs

If v and w are vertices of a graph F and e = v w] is an edge of Fthen we say that e joins v and w and that v and w are the ends of eThe number of edges of which v is an end is called the degree of v Asubgraph of F is constructed by taking a subset S of ET together withall vertices incident in F with some edge belonging to S An inducedsubgraph of F is obtained by taking a subset U of VT together withall edges which are incident in F only with vertices belonging to U Inboth cases the incidence relation in the subgraph is inherited from theincidence relation in F We shall use the notation (S)r (U)r for thesesubgraphs and usually when the context is clear the subscript F willbe omitted

PART ONE

Linear algebra in graph theory

The spectrum of a graph

We begin by defining a matrix which will play an important role in manyparts of this book Suppose that F is a graph whose vertex-set VT isthe set viV2-vn and consider ET as a set of unordered pairs ofelements of VT If viVj is in ET then we say that Vi and Vj areadjacent

Definition 21 The adjacency matrix of F is the n x n matrix A mdashA(F) whose entries a^ are given by

if Vi and Vj are adjacentotherwiseI1

toFor the sake of definiteness we consider A as a matrix over the complex

field Of course it follows directly from the definition that A is a realsymmetric matrix and that the trace of A is zero Since the rows andcolumns of A correspond to an arbitrary labelling of the vertices ofF it is clear that we shall be interested primarily in those propertiesof the adjacency matrix which are invariant under permutations of therows and columns Foremost among such properties are the spectralproperties of A

Suppose that A is an eigenvalue of A Then since A is real and sym-metric it follows that A is real and the multiplicity of A as a root ofthe equation det(AI mdash A) = 0 is equal to the dimension of the space ofeigenvectors corresponding to A

8 Linear algebra in graph theory

Definition 22 The spectrum of a graph F is the set of numbers whichare eigenvalues of A(F) together with their multiplicities If the distincteigenvalues of A(F) are Ao gt Aj gt gt As_i and their multiplicitiesare m(Ao)m(Ai) m(As_) then we shall write

A deg A l bull A s ~ JSpecF= (m(X0) m(Ai)

For example the complete graph Kn is the graph with n vertices inwhich each distinct pair are adjacent Thus the graph K4 has adjacencymatrix

A =

and an easy calculation shows that the spectrum of K4 is

3 - 1

-011

1

1011

1101

1-110

Spec K4 = bdquo

We shall usually refer to the eigenvalues of A = A(F) as the eigenval-ues ofT Also the characteristic polynomial det(AI mdashA) will be referredto as the characteristic polynomial oF and denoted by x(F A) Let ussuppose that the characteristic polynomial of F is

X(F A) = An + dA1 + c2A~2 + C3A-3 + + cn

In this form we know that mdash c is the sum of the zeros that is the sumof the eigenvalues This is also the trace of A which as we have alreadynoted is zero Thus c mdash 0 More generally it is proved in the theoryof matrices that all the coefficients can be expressed in terms of theprincipal minors of A where a principal minor is the determinant of asubmatrix obtained by taking a subset of the rows and the same subsetof the columns This leads to the following simple result

Proposition 23 The coefficients of the characteristic polynomial ofa graph F satisfy

(1) ci = 0(2) mdash c-i is the number of edges of F(3) mdashC3 is twice the number of triangles in F

Proof For each i s 12 n the number (mdashl)Ci is the sum ofthose principal minors of A which have i rows and columns So we canargue as follows

(1) Since the diagonal elements of A are all zero c = 0(2) A principal minor with two rows and columns and which has a


non-zero entry must be of the form0 11 0

There is one such minor for each pair of adjacent vertices of F and eachhas value mdash1 Hence (-l)2c2 = mdash ET giving the result

(3) There are essentially three possibilities for non-trivial principalminors with three rows and columns

010

100

000

5

011

100

100

011

101

110

and of these the only non-zero one is the last (whose value is 2) Thisprincipal minor corresponds to three mutually adjacent vertices in Fand so we have the required description of C3 bull

These simple results indicate that the characteristic polynomial of agraph is an object of the kind we study in algebraic graph theory it isan algebraic construction which contains graphical information Propo-sition 23 is just a pointer and we shall obtain a more comprehensiveresult on the coefficients of the characteristic polynomial in Chapter 7

Suppose A is the adjacency matrix of a graph F Then the set ofpolynomials in A with complex coefficients forms an algebra underthe usual matrix operations This algebra has finite dimension as acomplex vector space Indeed the Cayley-Hamilton theorem assertsthat A satisfies its own characteristic equation so the dimension is atmost n the number of vertices in F

Definition 24 The adjacency algebra of a graph F is the algebra ofpolynomials in the adjacency matrix A = A(F) We shall denote theadjacency algebra of F by A(T)

Since every element of the adjacency algebra is a linear combinationof powers of A we can obtain results about -4(F) from a study of thesepowers We define a walk of length I in F from Vi to Vj to be a finitesequence of vertices of F

such that Ut_i and

Vi = U0UiUi = Vj

are adjacent for 1 lt t lt

Lemma 25 The number of walks of length I in F from Vi to Vj isthe entry in position (i j) of the matrix A1

Proof The result is true for I = 0 (since A0 = I) and for = 1 (sinceA1 = A is the adjacency matrix) Suppose that the result is true for = L The set of walks of length L + 1 from Vi to Vj is in bijective


correspondence with the set of walks of length L from Vi to vertices Vhadjacent to Vj Thus the number of such walks is

J2 (A L k = JT(AL)ihahj = AL+l)i3vhvjeuro ET h=l

It follows that the number of walks of length L + 1 joining Vi to Vj is(AL+1)ij The general result follows by induction bull

A graph is said to be connected if each pair of vertices is joined bya walk The number of edges traversed in the shortest walk joining vând Vj is called the distance in F between vt and Vj and is denoted byd(viVj) The maximum value of the distance function in a connectedgraph F is called the diameter of F

Proposition 26 Let F be a connected graph with adjacency algebraA(T) and diameter d Then the dimension of AT) is at least d+1Proof Let x and y be vertices of F such that d(x y) = d and supposethat

x = wowiWd = y

is a walk of length d Then for each i 6 12 d there is at least onewalk of length i but no shorter walk joining WQ to Wi ConsequentlyA has a non-zero entry in a position where the corresponding entries ofI A A 2 A11 are zero It follows that A is not linearly dependenton I A A^1 and that I A Ad is a linearly independentset in A(T) Since this set has d+1 members the proposition is proved

bullThere is a close connection between the adjacency algebra and the

spectrum of F If the adjacency matrix has s distinct eigenvalues thensince it is a real symmetric matrix its minimum polynomial (the monicpolynomial of least degree which annihilates it) has degree s Conse-quently the dimension of the adjacency algebra is equal to s Thus wehave the following bound for the number of distinct eigenvalues

Corollary 27 A connected graph with diameter d has at least d+1distinct eigenvalues bull

One of the major topics of the last part of this book is the study ofa class of highly regular connected graphs which have the minimumnumber d + 1 of distinct eigenvalues In the following chapters we shallencounter several other examples of the link between structural regular-ity and the spectrum

The spectrum of a graph 11

Notation The eigenvalues of a graph may be be listed in two ways instrictly decreasing order of the distinct values as in Definition 22 or inweakly decreasing order (with repeated values) Ao gt Aj gt gt An_iwhere n = |VT| We shall use either method as appropriate

Additional Results

2a A reduction formula for x Suppose F is a graph with a vertexvi of degree 1 and let V2 be the vertex adjacent to v Let T bethe induced subgraph obtained by removing v and Fi2 the inducedsubgraph obtained by removing ui^- Then

This formula can be used to calculate the characteristic polynomial ofany tree because a tree always has a vertex of degree 1 A more generalreduction formula was found by Rowlinson (1987)

2b The characteristic polynomial of a path Let Pn be the path graphwith vertex-set vi V2 bull bull bull vn and edges vi Vi+i (1 lt i lt n mdash 1) Forn gt 3 we have

X(Pn A) = AX(Pn-i A) - X(Pn_2 A)

Hence x(Pni A) = Un(X2) where Un denotes the Chebyshev polynomialof the second kind

2c The spectrum of a bipartite graph A graph is bipartite if its vertex-set can be partitioned into two parts and V such that each edge hasone vertex in Vi and one vertex in V-i- If we order the vertices so thatthose in V come first then the adjacency matrix of a bipartite graphtakes the form

I B

If x is an eigenvector corresponding to the eigenvalue A and x is obtainedfrom x by changing the signs of the entries corresponding to vertices inV2 then x is an eigenvector corresponding to the eigenvalue mdashA Itfollows that the spectrum of a bipartite graph is symmetric with respectto 0 a result originally obtained by Coulson and Rushbrooke (1940) inthe context of theoretical chemistry

2d The derivative of Fdegr i = 1 2 n let Fj denote the inducedsubgraph (VTlaquoi) Then


2e The eigenvalue 0 Suppose that a graph has two vertices Vi and Vjsuch that the set of vertices adjacent to v^ is the same as the set of ver-tices adjacent to Vj Then the vector x whose only non-zero componentsare Xi = 1 and Xj = mdash1 is an eigenvector of the adjacency matrix witheigenvalue 0 If F has a set of r vertices all of which have the same setof neighbours then the multiplicity of 0 is at least r mdash 1 (An alternativeargument uses the observation that there are r equal columns of A andso its rank is at most n mdash r + 1)

2f Cospectral graphs Two non-isomorphic graphs are said to be cospec-tral if they have the same eigenvalues with the same multiplicities Thefirst example of this phenomenon was given by Collatz and Sinogowitz(1957) and many examples are given in [CvDS pp 156-161] Two con-nected graphs with 6 vertices both having the characteristic polynomialA6 - 7A4 - 4A3 + 7A2 + 4A - 1 are shown in Figure 1

Figure 1 two cospectral graphs

2g The walk-generating matrix Let gij (r) denote the number of walksof length r in F from Vi to Vj If we write G(z) for the matrix

r = l

then G(z) = (I mdash zA) l where A is the adjacency matrix of F Thismay be regarded as a matrix over the ring of formal power series in zor as a real matrix defined whenever z fi SpecF From the formula forthe inverse matrix and 2e we obtain

trG(z) =

2h Closed walks and sums of powers of eigenvalues A closed walk isone whose initial and final vertices coincide By Lemma 25 the totalnumber of closed walks of length I is equal to tr A Since the trace of amatrix is the sum of its eigenvalues an alternative expression is Yl^H-In particular the sum of the eigenvalues is zero the sum of the squares


is twice the number of edges and the sum of the cubes is six times thenumber of triangles

2i An upper bound for the largest eigenvalue Suppose that the eigen-values of F are Ao gt Ai gt gt An_i where F has n vertices and medges Prom 2h we obtain 52 A = 0 and J2 ^t = ^rn- I follows that

Another bound of the same type is Ao lt Z2ni mdash n + 1 (Yuan 1988)

2j The spectral decomposition (Godsil and Mohar 1988) The adjacencymatrix has a spectral decomposition A = 52âEa where the matricesEa are idempotent and mutually orthogonal It is easy to check thatgiven a set of mutually orthonormal eigenvectors xa we can take

Ea = xax that is (Ea)ij = (xa)j(xa)j

It follows that if is any function for which (A) is defined then (A) =]P (Aa)Ea For example the walk-generating matrix G(z) mdash ( I -Â) 1

is defined whenever z $ SpecF and it can be expressed in the form

a=0

This yields the following expression for the individual walk-generatingfunctions

n-l

a=0

2k The distance matrices For a graph with diameter d the distancematrices Aj (0 lt h lt d) are defined as follows

(A )bullbull = bull[ 1 if reg(Vu Vj^ ~ h]3 0 otherwise

It follows that

Ao = I Ai = A Ao + Ai + A2 + + Ad = Jwhere J is the matrix in which each entry is 1 The distance matrixAh can be expressed as a polynomial of degree h in A for each h in01 d if and only if the graph is distance-regular (see Chapter20) For such a graph the adjacency algebra has the minimum possibledimension d+1

Regular graphs and line graphs

In this chapter we discuss graphs which possess some kinds of combi-natorial regularity and whose spectra in consequence have distinctivefeatures A graph is said to be regular of degree k (or k-regular) if each ofits vertices has degree k This is the most obvious kind of combinatorialregularity and it has interesting consequences for the eigenvalues

Proposition 31 Let T be a regular graph of degree k Then(1) k is an eigenvalue oF(2) ifT is connected then the multiplicity of k is 1(3) for any eigenvalue A ofT we have X lt k

Proof (1) Let u = [ 1 1 1] then if A is the adjacency matrix ofF we have Au = fcu since there are k ls in each row Thus fc is aneigenvalue of F

(2) Let x = [aJiX2an] denote any non-zero vector for whichAx = fcx and suppose that Xj is an entry of x with the largest absolutevalue Since (Ax)j = kxj we have

ZJ X$ ~-mdash KXj

where pound denotes summation over those k vertices Vi which are adjacentto Vj By the maximal property of Xj it follows that xt = Xj for allthese vertices If F is connected we may proceed successively in this wayeventually showing that all entries of x are equal Thus x is a multipleof u and the space of eigenvectors associated with the eigenvalue k hasdimension 1

(3) Suppose that Ay = Ay y ^ 0 and let yj denote an entry of y

Regular graphs and line graphs 15

which is largest in absolute value By the same argument as in (2) wehave Syi = Xyj and so

yj = XyiltVyiltkyj

Thus |A| lt k as required bull

The adjacency algebra of a regular connected graph also has a distinc-tive property related to the results of Proposition 31 Let J denote thematrix each of whose entries is +1 Then if A is the adjacency matrixof a regular graph of degree k we have AJ = JA = kJ This is the pointof departure for the following result

Proposition 32 (Hoffman 1963) The matrix J belongs to the adja-cency algebra A(T) if and only if T is a regular connected graph

Proof Suppose J is in A(T) By the definition of A(T) J is a polyno-mial in A consequently AJ = JA Now if k^ denotes the degree of thevertex vit then (AJ)^ = k^ and (AJ) = k^ so that all the degreesare equal and F is regular Further if F were disconnected we couldfind two vertices with no walks joining them so that the correspondingentry of A1 would be zero for all I gt 0 Then every polynomial in Awould have a zero entry contradicting the fact that J euro A(F) Thus Fis connected

Conversely suppose that F is connected and regular of degree k Thenby part (1) of Proposition 31 k is an eigenvalue of F and so the mini-mum polynomial of A is of the form p) = (A mdash k)q) Since pA) = 0we obtain Aq(A) = kq(A) that is each column of q(A) is an eigenvec-tor of A corresponding to the eigenvalue k By part (2) of Proposition31 it follows that each column of q(A) is a multiple of u and since q(A)is a symmetric matrix it is a multiple of J Thus J is a polynomial inA bull

Corollary 33 Let T be a k-regular connected graph with n verticesand let the distinct eigenvalues ofT be k gt Xi gt gt As_i Then ifq(X) = ]J(X - Xi) where the product is over the range 1 lt i lt s - 1 wehave

Proof It follows from the proof of Proposition 32 that q(A) = aJfor some constant a Now the eigenvalues of q(A) are q(k) and q(Xi) for1 lt i lt s mdash 1 and all of these except q(k) are zero The only non-zeroeigenvalue of aJ is an hence a = q(k)n bull


For some classes of regular graphs such as the strongly regular graphs(3c) it is possible to determine the polynomial function for which(A) = J by direct means based on Lemma 25 This provides a pow-erful method for determining the spectra of these graphs At a morebasic level there is a special class of regular graphs whose spectra canbe found by means of a well-known technique in matrix theory Asthis class contains several important families of graphs we shall brieflyreview the relevant theory

An nxn matrix S is said to be a circulant matrix if its entries satisfySij mdash Sij-i+i where the subscripts are reduced modulo n and lie in theset 12 n In other words row i of S is obtained from the firstrow of S by a cyclic shift of i mdash 1 steps and so any circulant matrix isdetermined by its first row Let W denote the circulant matrix whosefirst row is [010 0] and let S denote a general circulant matrixwhose first row is [siS2sn] Then a straightforward calculationshows that

Since the eigenvalues of W are 1 w w2 un~1 where ui = exp(27rin)it follows that the eigenvalues of S are

Definition 34 A circulant graph is a graph T whose vertices can beordered so that the adjacency matrix A(F) is a circulant matrix

The adjacency matrix is a symmetric matrix with zero entries on themain diagonal It follows that if the first row of the adjacency matrixof a circulant graph is [a 02an] then a mdash 0 and o = an-i+2 fori-2n

Proposition 35 Suppose that [0d2 bull bull bull an] is the first row of theadjacency matrix of a circulant graph F Then the eigenvalues ofT are

3=2

Proof This result follows directly from the expression for the eigen-values of a circulant matrix bull

We remark that the n eigenvalues given by the formula of Proposition35 are not necessarily all distinct


We shall give three examples of this technique First the completegraph Kn is a circulant graph the first row of its adjacency matrix is[0 l l l ] Since

1 + ujr + + o(n~1)r = 0 for r euro 12 n - 1

it follows from Proposition 35 that the spectrum of Kn is

Spec _(n- - 1 Kn~ 1 n-l)-

1 J

Our second example is the cycle graph Cn whose adjacency matrixis a circulant matrix with first row [010 01] In the notationof Proposition 35 the eigenvalues are Ar = 2cos(2rrn) but thesenumbers are not all distinct taking account of coincidences the completedescription of the spectrum is

c n _ (^ ^cos 2Kjri 2 cos(n mdashb p e c C n - ^ j 2 2

n (2 2cos27rn 2cos(n-2)bpec Ln mdash I 1 o oVI Z L

A third family of circulant graphs are the graphs Hs obtained by re-moving s disjoint edges from K2S- The graph Hs is sometimes known asa hyperoctahedral graph because it is the skeleton of a hyperoctahedronin s dimensions It is also known as the cocktail-party graph CP(s)y socalled because it is alleged that if there are s married couples at a cock-tail party each person talks to everyone except their spouse It is alsoa special kind of complete multipartite graph to be defined in Chapter6 Clearly the graph Hs is a circulant graph we may take the first rowof its adjacency matrix to be [oi a2S] where each entry is 1 exceptthat a = as+i = 0 It follows that the eigenvalues of Hs are

Ao = 2 s - 2 Ar = - l - a r s (1 lt r lt 2s - 1)

where UJ2S = 1 and u) ^ 1 Consequently

s - 2 0 - 2

We now turn to another structural property which has implications forthe spectrum of a graph The line graph L(T) of a graph F is constructedby taking the edges of F as vertices of L(F) and joining two vertices inL(V) whenever the corresponding edges in F have a common vertexThe spectra of line graphs were investigated extensively by Hoffman(1969) and others Here we outline the basic results more recent workis described in the Additional Results at the end of the chapter


We shall continue to suppose that F has n vertices VV2- bull vn Weshall need to label the edges of F also that is ET = e e^ em Forthe purposes of this chapter only we define a n n x m matrix X = X(F)as follows

and ej are incidentotherwise

f l Xvi 0 othe

Lemma 36 Suppose that F and X are as above Let A denote theadjacency matrix ofT and AL the adjacency matrix of L(T) Then

(1) XX = AL + 2Im(2) if F is regular of degree k then XX = A + kln

The subscripts denote the sizes of the identity matrices

Proof (1) We have

from which it follows that (XX)jj is the number of vertices vi of Fwhich are incident with both the edges e and ej The required result isnow a consequence of the definitions of L(T) and A^

(2) This part is proved by a similar counting argument D

Proposition 37 A is an eigenvalue of a line graph L(T) thenAgt - 2

Proof The matrix XX is non-negative definite since we have zXXz= ||Xz||2 gt 0 for any vector z Thus the eigenvalues of XX are non-negative But Ai = XX mdash 2Im so the eigenvalues of AL are not lessthan - 2 bull

The condition that all eigenvalues of a graph be not less than mdash2is a restrictive one but it is not sufficient to characterize line graphsFor example the hyperoctahedral graphs Hs satisfy this condition butthese graphs are not line graphs Seidel (1968 see 3g) gave examplesof regular graphs which have least eigenvalue mdash2 and are neither linegraphs nor hyperoctahedral graphs Subsequently a characterization ofall graphs with least eigenvalue mdash2 was obtained by Cameron GoethalsSeidel and Shult (1976 see 3i)

When F is a regular graph of degree k its line graph L(T) is regularof degree 2c mdash 2 We can think of this as a connection between themaximum eigenvalues of F and L(T) and in fact the connection extendsto all eigenvalues by virtue of the following result


Theorem 38 (Sachs 1967) IfT is a regular graph of degree k with nvertices and m = |nfc edges then

X(L(T) A) = (A + 2)m-nX(T X + 2-k)

Proof We shall use the notation and results of Lemma 36 Definetwo partitioned matrices with n + m rows and columns as follows

TT _ AIn mdashX _ _ Ira XL U l m J [A Aim

Then we haveAIlaquo-XX 0 ] TAIn 0

J LT T V mdash trade ---ltraquobull ATT mdash[ -ltv A l m J [ AJV Aljn - A A

Since det(UV) = det(VU) we deduce that

Amdet(AIn - XX) = Adet(AIm - XX)

Thus we may argue as follows

x (pound ( r ) A)=de t (AI m -A L )

= det((A + 2)Im - XX)

= (A + 2)m-det((A + 2)In - XX4)

= (A + 2)m-det((A + 2- jfe)In - A)

D

It follows from Theorem 38 that if the spectrum of T is

k X As_Spec r =

then the spectrum of L(T) is

Spec LCT) = 1 1 mi ms-i m mdash n

For example the line graph L(Kt) is sometimes called the trianglegraph and denoted by At Its vertices correspond to the ^t(t mdash 1) pairs ofnumbers from the set 12 t two vertices being adjacent wheneverthe corresponding pairs have just one common member From the knownspectrum of Kt and Theorem 38 we have

2 t - 4 i - 4 - 2


Additional results

3a The complement of a regular graph Let F be a graph with n verticesand let Fc denote its complement that is the graph with the samevertex-set whose edge-set is complementary to that of F Let Ac denotethe adjacency matrix of Fc Then A + Ac = J mdash I It was proved bySachs (1962) that if F is connected and regular of degree fc then

(A + fc + 1)X(FC A) - (-l)n(A - n + k + 1)X(F -A - 1)

3b The Petersen graph The complement of the line graph of K5 isknown as the Petersen graph It occurs in many contexts throughoutgraph theory We shall denote it by the symbol O3 as it is the casek = 3 of the family Ok of odd graphs to be defined later (8f) Wehave

3 1 - 2 s

Spec O3 = 1 g

In particular the least eigenvalue is mdash2 although O3 is neither a linegraph nor a hyperoctahedral graph

3c Strongly regular graphs A fc-regular graph is said to be stronglyregular with parameters (k a c) if the following conditions hold Eachpair of adjacent vertices has the same number a gt 0 of common neigh-bours and each pair of non-adjacent vertices has the same number c gt 1of common neighbours It follows from Lemma 25 that the adjacencymatrix of such a graph satisfies

A2 + (c - o)A + (c - fc)I = cJ

In other words the polynomial function whose existence is guaranteedby Proposition 32 is fx) = (lc)(a2 + (c - a)x + (c - A))

3d The spectrum of a strongly regular graph Since the eigenvalues ofthe n x n matrix J are n (with multiplicity 1) and 0 (with multiplicitynmdash 1) it follows from 3c that the eigenvalues of a strongly regular graphare k (with multiplicity 1) and the two roots Ai A2 of the quadraticequation (A) = 0 (with total multiplicity n mdash 1) The multiplicitiesmi = m(Ai) and m^ mdash m^2) can be determined from the equations

m + mi = n mdash 1 k + miAi + TO2A2 = 0

the second of which follows from 2h For example the Petersen graph(3b) is strongly regular with parameters (301) and this gives an al-ternative method of determining its spectrum

3e The Mobius ladders The Mobius ladder M^ is a regular graph ofdegree 3 with 2h vertices (h gt 3) It is constructed from the cycle graph


Cih by adding new edges joining each pair of opposite vertices and soit is a circulant graph The eigenvalues are the numbers

Xj = 2 COS(TT jh) + ( - l ) J (0ltjlt2h- 1)

3f Graphs characterized by their spectra Although there are many ex-

amples of cospectral graphs there are also cases where there is a uniquegraph with a given spectrum We give two instances

(o) The spectrum of the triangle graph At = L(Kt) is given above IfF is a graph for which SpecF = SpecA^ and t ^ 8 then F = At Inthe case t = 8 there are three exceptional graphs not isomorphic withAs but having the same spectrum as As (Chang 1959 Hoffman 1960)

(b) The complete bipartite graph Kalta is constructed by taking twosets of a vertices and joining every vertex in the first set to every vertexin the second If F is a graph for which SpecF = Speci(Xa i O) anda 7 4 then F = LKaâ) In the case a = 4 there is one exceptionalgraph this graph is depicted in Figure 2 (Shrikhande 1959)

13

Figure 2 Shrikhandes exceptional graph

3g Regular graphs with least eigenvalue - 2 The following graphs hav-ing least eigenvalue mdash2 were noted by Seidel (1968) They are neitherline graphs nor hyperoctahedral graphs

(a) the Petersen graph(b) a 5-regular graph with 16 vertices(c) a 16-regular graph with 27 vertices (see p 57)(d) the exceptional graphs mentioned in 3f

3h Generalized line graphs The cocktail party graph CP(s) is definedon page 17 For any graph F with vertices viV2--vn and any non-negative integers a a-i an we construct the generalized line graphL(T aia2 bull bull an) as follows The vertex-set is the union of the vertex-sets of L(F) CP(ai ) CP(a2) bull bull bull CP(an) and the edge-set is the union


of the edge-sets together with edges joining all vertices of CP(ai) toevery vertex of L(T) corresponding to an edge of F containing Vi for1 lt i lt n A generalized line graph constructed in this way has leasteigenvalue mdash2

3i All graphs with least eigenvalue mdash2 If F is a graph with least eigen-value not less than mdash2 then A + 21 is non-negative definite and soA + 21 = MM for some matrix M By establishing a correspon-dence between the rows of M and sets of vectors known as root systemsCameron Goethals Seidel and Shult (1976) showed that all graphswith least eigenvalue not less than mdash2 fall into three classes (a) the linegraphs of bipartite graphs (b) the generalized line graphs described in3h (c) a finite class of graphs arising from the root systems E^ Er E$

3j Perfect codes in regular graphs For any vertex v of a graph F definethe e-neighbourhood of v to be Ne(v) = u euro VT | d(u v) lt e Aperfect e-code in F is a set C C VT such that the e-neighbourhoodsNe(c) with c euro C form a partition of VT Suppose that C is a perfect1-code in a ^-regular graph F Then the vector c which takes the value1 on vertices in C and 0 on other vertices satisfies Ac = u mdash c It followsthat u mdash (k + l)c is an eigenvector of F with eigenvalue mdash1 Thus anecessary condition for a regular graph to have a perfect 1-code is thatmdash1 is an eigenvalue (See also 21j)

3k Spectral bounds for the diameter Suppose that T is connected andA-regular so that XQ = k and Ai lt k Alon and Milman (1985) provedthat the diameter d is bounded above by a function of n k and thegap k mdash specifically

d lt 2 2k V log2n

Mohar (1991) improved this to

Using the spectral decomposition of Ar (see 2j) Chung (1989) obtaineda bound involving the second largest eigenvalue in absolute value A =max(Ai -An_i) She showed that if (kA)r gt n mdash 1 then (Ar)f bull gt 0for all ij It follows that

ln(n-l)d lt

Cycles and cuts

Let C denote the field of complex numbers and let X be any finite setThen the set of all functions from X to C has the structure of a finite-dimensional vector space if X mdashgt C and g X mdashgtbull ltC then the vectorspace operations are defined by the rules

( + 9)(x) = fx) + g(x) (a)(x) = af(x) (xGXaeuro C)

The dimension of this vector space is equal to the number of membersof X

Definition 41 The vertex-space Cb(F) of a graph is the vector spaceof all functions from VT to C The edge-space C (F) of F is the vectorspace of all functions from ET to C

Taking VT = viv2 vn and ET = eie2 em it followsthat Co(F) is a vector space of dimension n and Ci(F) is a vector spaceof dimension m Any function r VT mdashbull C can be represented by acolumn vector

y = [yiy2---ynt

where jj = r(vi) (1 lt i lt n) This representation corresponds to choos-ing as a basis for Co(F) the set of functions wiogt2- -wn definedby

i _ 1) if = j UiVj)~0 otherwise

In a similar way we may choose the basis eje2 em fdegr Ci(F)


defined by

0 otherwiseand hence represent a function pound ET -raquo C by a column vector x =[xiX2---xm]t such that xraquo = pound(eraquo) (1 lt i lt m) We shall refer tothe bases ugtiu2 wn and ei pound2 em as the standard bases forCo(r)andC(r)

We now introduce a useful device For each edge ea = vrrvT of Fwe shall choose one of vavT to be the positive end of ea and the otherone to be the negative end We refer to this procedure by saying that Fhas been given an orientation Although this device is employed in theproofs of several results the results themselves are independent of it

Definition 42 The incidence matrix D of F with respect to a givenorientation of F is the n xm matrix (d^) whose entries are

+1 if Vi is the positive end of e mdash 1 if Vi is the negative end of ej0 otherwise

The rows of the incidence matrix correspond to the vertices of F andits columns correspond to the edges of F each column contains just twonon-zero entries +1 and mdash 1 representing the positive and negative endsof the corresponding edge

We remark that D is the representation with respect to the standardbases of a linear mapping from Ci(F) to CQ(T) This mapping will becalled the incidence mapping and be denoted by D For each pound ET mdashgt Cthe function Df VT mdash C is defined by

For the rest of this chapter we shall let c denote the number of con-nected components of F

Proposition 43 The incidence matrix D o F has rank n mdash cProof The incidence matrix can be written in the partitioned form

D(1) 0 00 Dlt2gt 0

0 0 D^c

by a suitable labelling of the vertices and edges of F where the matrixpoundgt() (1 lt i lt c) is the incidence matrix of a component fM of F Weshall show that the rank of D^-1 is n mdash 1 where rij = |VT^| from whichthe required result follows by addition

Cycles and cuts 25

Let dj denote the row of D ^ corresponding to the vertex Vj of T^Since there is just one +1 and just one mdash1 in each column of D1 itfollows that the sum of the rows of D1 is the zero row vector and thatthe rank of D^ is at most rii mdash Suppose we have a linear relation^Zctjdj = 0 where the summation is over all rows of D ^ and not allthe coefficients otj are zero Choose a row dk for which a^ ^ 0 thisrow has non-zero entries in those columns corresponding to the edgesincident with Vk- For each such column there is just one other row d

with a non-zero entry in that column and in order that the given linearrelation should hold we must have a = a^ Thus if at ^ 0 thena = ajt for all vertices vi adjacent to v^ Since F ^ is connected itfollows that all coefficients OLJ are equal and so the given linear relationis just a multiple of J^ dj = 0 Consequently the rank of D ^ is rii mdash 1

bullThe following definition applies to a general graph F with n vertices m

edges and c components although for the time being we shall continueto deal with strict graphs rather than general graphs

Definition 44 The rank of F and the co-rank of F are respectively

r(F) = n mdash c s(F) = m mdash n + c

We now investigate the kernel of the incidence mapping D and itsrelationship with graph-theoretical properties of F Let Q be a set ofedges such that the subgraph Q) is a cycle graph We say that Q is acycle in F the two possible cyclic orderings of the vertices of (Q) inducetwo possible cycle-orientations of the edges Q Let us choose one of thesecycle-orientations and define a function poundQ in C (F) as follows We putpoundq(e) = +1 if e belongs to Q and its cycle-orientation coincides with itsorientation in F cj(e) = mdash 1 if e belongs to Q and its cycle-orientation isthe reverse of its orientation in F while if e is not in Q we put ^Q(e) = 0

Theorem 45 The kernel of the incidence mapping D ofT is a vectorspace whose dimension is equal to the co-rank oF If Q is a cycle in Fthen poundQ belongs to the kernel of D

Proof Since the rank of D is n mdash c and the dimension of C (F) is mit follows that the kernel of D has dimension m mdash n + c = s(F) Withrespect to the standard bases for Ci(F) and Co(F) we may take D to bethe incidence matrix and poundQ to be represented by a column vector XQNow (DXQ)J is the inner product of the row d of D and the vector XQ

If vt is not incident with some edges of Q then this inner product is 0if Vi is incident with some edges of Q then it is incident with precisely


two edges and the choice of signs in the definition of poundQ implies that theinner product is again 0 Thus Dxg = 0 and poundQ belongs to the kernelof D U

If p and a are two elements of the edge-space of F (that is functionsfrom ET to ltC) then we may define their inner product

where the over line indicates the complex conjugate When p and a arerepresented by coordinate vectors with respect to the standard basisof Ci(F) this inner product corresponds to the usual inner productof vectors in the complex vector space Cm (In practice we use onlyfunctions with real values so the conjugation is irrelevant)

Definition 46 The cycle-subspace of F is the kernel of the incidencemapping of F The cut-subspace of F is the orthogonal complement ofthe cycle-subspace in Ci(F) with respect to the inner product definedabove

The first part of this definition is justified by the result of Theorem 45which says that vectors representing cycles belong to the cycle-subspaceindeed in the next chapter we shall show how to construct a basis for thecycle-subspace consisting entirely of cycles We now proceed to justifythe second part of the definition

Let VT = V U V2 be a partition of VT into non-empty disjoint subsetsIf the set H of edges of F which have one vertex in Vj and one vertex inV2 is non-empty then we say that if is a cut in F We may choose one ofthe two possible cut-orientations for H by specifying that one of Vi V2

contains the positive ends of all edges in H while the other containsthe negative ends We now define a function poundH in Ci(F) by puttingpoundtf (e) = +1 if e belongs to H and its cut-orientation coincides with itsorientation in F pound (e) = mdash 1 if e belongs to H and its cut-orientation isthe reverse of its orientation in F and pound(e) = 0 if e is not in H

Proposition 47 The cut-subspace of F is a vector space whose di-mension is equal to the rank ofT If H is a cut in F then pound belongsto the cut-subspace

Proof Since the dimension of the cycle-subspace is m mdash n 4- c itsorthogonal complement the cut-subspace has dimension n mdash c = r(F)

If if is a cut in F we have VT mdash Vi U V2 where V and V2 are disjointand non-empty and H consists precisely of those edges which have onevertex in V and one vertex in V2- Thus if x is the column vector

Cycles and cuts 27

representing pound we have

where dj is the row of the incidence matrix corresponding to Vi Thesign on the right-hand side of this equation depends only on which ofthe two possible cut-orientations has been chosen for H Now if Dz = 0then djZ = 0 for each v^ pound V and we deduce that x^z = 0 In otherwords poundH belongs to the orthogonal complement of the cycle-subspaceand by definition this is the cut-subspace D

The proof of Proposition 47 indicates one way of choosing a basispoundiipound2j bull bull bull poundn-c for the cut-subspace of F The set of edges incidentwith a vertex Vj of F forms a cut whose representative vector is d-If for each component F ^ (1 lt i lt c) of F we delete one row of Dcorresponding to a vertex in T^ then the remaining n - c rows arelinearly independent Furthermore the transpose of any vector xHrepresenting a cut H can be expressed as a linear combination of thesen mdash c rows by using the equation displayed in the proof of Proposition47 and the fact that the sum of rows corresponding to each componentisO

This basis has the desirable property that each member represents anactual cut rather than a linear combination of cuts It is howeverrather clumsy to work with and in the next chapter we shall investigatea more elegant procedure which has the added advantage that it providesa basis for the cycle-subspace as well

We end this chapter by proving a simple relationship between theLaplacian matrix Q = DD and the adjacency matrix of F

Proposition 48 Let D be the incidence matrix (with respect to someorientation) of a graph F and let A be the adjacency matrix ofT Thenthe Laplacian matrix Q satisifies

Q = DD = A - A

where A is the diagonal matrix whose ith diagonal entry is the degreeof the vertex V (1 lt i lt n) Consequently Q is independent of theorientation given to F

Proof (DD)jj is the inner product of the rows d and dj of D Ifi 7 j then these rows have a non-zero entry in the same column if andonly if there is an edge joining u and Vj In this case the two non-zeroentries are +1 and - 1 so that (DD- = - 1 Similarly (DD^ is theinner product of dj with itself and since the number of entries plusmn1 indi is equal to the degree of vt the result follows D


Additional Results

4a The coboundary mapping The linear mapping from Co(F) to Ci(F)defined (with respect to the standard bases) by x gt-raquo Dx is sometimescalled the coboundary mapping for F The kernel of the coboundary map-ping is a vector space of dimension c and the image of the coboundarymapping is the cut-subspace of F

4b The isoperimetric number For any set X C VT the cut defined bythe partition of VT into X and its complement is denoted by 6X Theisoperimetric number of F is defined to be

i(T) = min T ^ T -|X|lt|vr|2 X

For example it is easy to check that i(Kn) = [n-2] i(03) = 1

4c Small cycles The girth of a graph is the number g of edges in asmallest cycle For example g(Kn) = 3 (n gt 3) g(Kaa) mdash 4 (a gt 2)and 5(03) = 5 If F has girth g gt 2r + 1 then for each pair of verticesv and w such that d(v w) = q lt r there is a unique walk of length qfrom v to w In the A-regular case this leads to the following relationsbetween the adjacency matrix and the distance matrices A (2 lt q lt r)defined in 2k

A2 = A2 - fcl Aq = AA_ - (fc - 1)A_2 (3 lt q lt r)

It follows that a distance matrix Aq with q lt r is expressible as apolynomial in A Explicitly Aq = fq(A) where

fo(x) = 1 fx(x)=x f2(x) = x2-k

fq(x) - xfq-xx) ~(k- l) -2(i) (q gt 3)

4d Girift and excess It is an elementary exercise (see Chapter 23)to show that the number of vertices in a fc-regular graph with girthg mdash 2r + 1 is at least

no(kg) = l + k + k(k-l)+k(k-l)2 + + k(k- l)r

The Petersen graph O3 achieves the lower bound no for the case k = 3and g = 5 but in the general case graphs which achieve the lower boundare rare (Chapter 23) For any fc-regular graph F with girth g we definethe excess to be the amount e by which the lower bound is exceededthat is e = n mdash no(k g) where n is the number of vertices in F Usingthe equations given in 4c Biggs (1980) established a lower bound for ein terms of the eigenvalues of F Define the polynomials gj by

9ix) = 0(a) + fix) + + fi(x)

Cycles and cuts 29

where the polynomials are defined above Then for any fc-regulargraph with girth g = 2r + 1 the excess e satisfies

egt|lt7r(A)| (A e Spec F X ^ k)

4e The Laplacian spectrum Let io lt î lt bull bull bull lt Mlaquo-i D e the eigen-values of the Laplacian matrix Q Then

(a) (JQ = 0 with eigenvector [ 1 1 1](b) if F is connected J gt 0(c) if F is regular of degree k then u = k - A where the Aj are

the (ordinary) eigenvalues of F in weakly decreasing order

4f Planar graphs and duality A planar graph is one which can bedrawn in the plane in the usual way without extraneous crossings of theedges The dual of a graph so drawn is the graph whose vertices arethe resulting regions of the plane two being adjacent when they have acommon edge Let F be a connected planar graph and F a dual of FIf F is given an orientation and D is the incidence matrix of F then Fcan be given an orientation so that its incidence matrix D satisfies

(a) rank (D) + rank (D) = |poundT|(b) DD = 0

4g The image of the incidence mapping Let w be an element of C0(F)where F is a connected graph Then ugt is in the image of D if and onlyif

vevrA more sophisticated way of expressing this result is as follows LetS Co(F) mdashgt C denote the linear map defined by S^w) = ^Zw(u) this isknown as the augmentation map Then the sequence of linear maps

Ci(r) - ^ co(F) -poundgt c mdashgt ois exact In particular this means that the image of D is equal to thekernel of 5

4h Flows An element ltj) of the cycle-subspace of F is called a flow on FThe support of ltf) written S(4gt) is the set of edges e for which ^(e) ^ 0a subset S of ET is a minimal support if S = S(ltfgt) for some flow 0 andthe only flow whose support is properly contained in S is the zero flowWe have the following basic facts

(a) The set of flows with a given minimal support (together withthe zero flow) forms a one-dimensional space


(b) A minimal support is a cycle(c) If ltfi is a flow whose support is minimal then |(gt(e)| is constant

on S((fgt)

4i Integral flows The flow 4gt is integral if each ltfi(e) is an integer it isprimitive if S(ltp) is minimal and each (jgt(e) is 01 or mdash1 We say that theflow 0 conforms to the flow if S(6) C S(x) and 8(e)x(e) gt 0 for e inS(6) Tutte (1956) showed that

(a) for a given integral flow ltfr there is a primitive flow which con-forms to ltfgt

(b) any integral flow ltjgt is the sum of integer multiples of primitiveflows each of which conforms to ltfgt

4j Modular flows Suppose the entries 01 mdash1 of D are taken to beelements of the ring TLU = Z u Z of residue classes of integers modulo uA flow mod u on F is a vector x with components in Z u for which Dx= 0 where 0 is the zero vector over Z u The results in 4i imply that ifx is a given flow mod u then there is an integral flow y each of whosecomponents jj satisfies y pound Xi and mdash u lt yi lt u Consequently if F hasa flow mod u then it has a flow mod (u + 1) (Tutte 1956)

4k The 5-flow conjecture A nowhere-zero f-flow ltjgt on T is a flow mod for which S(ltfgt) = ET Tutte (1954) conjectured that every graph withno isthmus has a nowhere-zero 5-flow (An isthmus is a cut consistingof a single edge) The following results are known

(a) Every planar graph with no isthmus has a nowhere-zero 4-flow(b) The Petersen graph does not have a nowhere-zero 4-flow(c) Every graph with no isthmus has a nowhere-zero 6-flow (Sey-

mour 1981)

Spanning trees and associated structures

The problem of finding bases for the cycle-subspace and the cut-subspaceis of great practical and theoretical importance It was originally solvedby Kirchhoff (1847) in his studies of electrical networks and we shallgive a brief exposition of that topic at the end of the chapter

We shall restrict our attention to connected graphs because the cycle-subspace and the cut-subspace of a disconnected graph are the directsums of the corresponding spaces for the components Throughout thischapter F will denote a connected graph with n vertices and m edgesso that r(F) = n mdash 1 and s(F) = m mdash n + 1 We shall also assume thatF has been given an orientation

A spanning tree in F is a subgraph which has n mdash 1 edges and containsno cycles It follows that a spanning tree is connected We shall use thesymbol T to denote both the spanning tree itself and its edge-set Thefollowing simple lemma is a direct consequence of the definition

Lemma 51 Let T be a spanning tree in a connected graph F Then(1) for each edge gofT which is not in T there is a unique cycle in F

containing g and edges in T only(2) for each edge h of T which is in T there is a unique cut in F

containing h and edges not in T only D

We write cyc(T g) and cut(T h) for the unique cycle and cut whoseexistence is guaranteed by Lemma 51 We give cyc(Tg) and cut(T h)the cycle-orientation and cut-orientation which coincide on g and hrespectively with the orientation in F Then we have elements


and poundTh) of the edge-space Ci(F) these elements axe defined (in termsof the given cycle and cut) as in Chapter 4

Theorem 52 With the same hypothesis as in Lemma 51 we have(1) as g runs through the set poundT mdash T the m mdash n + 1 elements pound(T9)

form a basis for the cycle-sub space of F(2) as h runs through the set T the n mdash 1 elements euro(Th) form a basis

for the cut-subspace of F

Proof (1) Since the elements pound(Tg) correspond to cycles it followsfrom Theorem 45 that they belong to the cycle-subspace They forma linearly independent set because a given edge g in EF mdash T belongsto cyc(T g) but to no other cyc(T g) for g ^ g Finally since thereare m - n + 1 of these elements and this is the dimension of the cycle-subspace it follows that we have a basis

(2) This is proved by arguments analogous to those used in the proofof the first part bull

We shall now put the foregoing ideas into a form which will showexplicitly how cycles and cuts can be derived from the incidence matrixby means of simple matrix operations To do this we shall require someproperties of submatrices of the incidence matrix

Proposition 53 (Poincare 1901) Any square submatrix of the inci-dence matrix D of a graph F has determinant equal to 0 or +1 or mdash 1

Proof Let S denote a square submatrix of D If every column of Shas two non-zero entries then these entries must be +1 and mdash1 and sosince each column has sum zero S is singular and det S = 0 Also ifevery column of S has no non-zero entries then det S = 0

The remaining case occurs when a column of S has precisely one non-zero entry In this case we can expand det S in terms of this columnobtaining det S = plusmn det S where S has one row and column fewerthan S Continuing this process we eventually arrive at either a zerodeterminant or a single entry of D and so the result is proved bull

Proposition 54 Let U be a subset of ET with U = n - 1 Let Tgtudenote an (nmdash 1) x (n mdash 1) submatrix ofTgt consisting of the intersectionof those n mdash 1 columns of D corresponding to the edges in U and any setof n mdash 1 rows of D Then Du is invertible if and only if the subgraphU) is a spanning tree ofT

Proof Suppose that (U) is a spanning tree of F Then the submatrixDy consists of n - 1 rows of the incidence matrix D of U Since (U) isconnected the rank of D is n mdash 1 and so Tgtu is invertible

Spanning trees and associated structures 33

Conversely suppose that Tgtu is invertible Then the incidence matrixD of (U) has an invertible (n mdash 1) x (n-1) submatrix and consequentlythe rank of D is (n mdash 1) Since U = n mdash 1 this means that the cycle-subspace of (U) has dimension zero and so (U) is a spanning tree of

r bull

Suppose that VT = v vlti bull bull bull vn and ET = e 62 bull bull bull em wherethe labelling has been chosen so that e elti en_i are the edges of agiven spanning tree T of T The incidence matrix of F is then partitionedas follows

where D T is an (n mdash 1) x (n mdash 1) square matrix invertible by Proposition54 and the last row dn is linearly dependent on the other rows

Let C denote the matrix whose columns are the vectors representingthe elements pound(Tej) (^ lt j lt m) with respect to the standard basis ofCi(F) Then C can be written in the partitioned form

CT 1c=

bulllm-n+1

Since every column of C represents a cycle and consequently belongs tothe kernel of D we have DC = 0 Thus

T mdash 1 T^T1 mdash mdashJLJrp bull y -

In a similar fashion the matrix K whose columns represent the elementspound(Tej) (1 lt 3 lt n ~ 1) c a n De written in the form

X-ir

Since each column of K belongs to the orthogonal complement of thecycle-subspace we have CK = 0 that is Cx + KT = 0 Thus

JX = (Uy UN) bull

Our equations for Cx and Kx show how the basic cycles and cuts asso-ciated with T can be deduced from the incidence matrix We also havean algebraic proof of the following proposition

Proposition 55 Let T be a spanning tree of T and let a and b beedges of T such that aeuroTbampT Then

b 6 cut(r a)ltae cyc(T b)

Proof This result follows immediately from the definitions of Cx andKT and the fact that C T + K^ = 0 bull


We end this chapter with a brief exposition of the solution of networkequations this application provided the stimulus for Kirchhoffs devel-opment of the foregoing theory in the middle of the nineteenth century

An electrical network is a connected graph F (with an arbitrary ori-entation) which has certain physical characteristics specified by twovectors in the edge-space of F These vectors are the current vector wand the voltage vector z These vectors are related by a linear equa-tion z = Mw + n where M is a diagonal matrix whose entries are theconductances of the edges and n represents externally applied voltagesFurther w and z satisfy the equations

Dw = 0 Cz = 0

which are known as Kirchhoffs laws If we choose a spanning tree T inF and partition D and C as before then the same partition on w andz gives

[ wj-1w = z =

Now from Dw = 0 we have D T W ^ + DJVWJV = 0 and since Cx =mdashD^Dy it follows that

wj- = CXWJV and w = CWJV-

In other words all the entries of the current vector are determined by theentries corresponding to edges not in T Substituting in z = Mw + nand premultiplying by C we obtain

(CtMC)wJV = -C n Since CMC is a square matrix with size and rank both equal to mmdashn+1it is invertible

So this equation determines WN and consequently both w (fromw = CWAT) and z (from z = Mw + n) in turn Thus we have a system-atic method of solving network equations which distinguishes clearlybetween the essential unknowns and the redundant ones

Additional Results

5a Total unimodularity A matrix is said to be totally unimodular if ev-ery square submatrix of it has determinant 0 1 or mdash1 thus Proposition53 states that D is totally unimodular A generalisation of this resultwas proved by Heller and Tompkins (1956) They showed that if M isa matrix with elements 0 1 or mdash1 such that every column contains atmost two non-zero elements then M is totally unimodular if and onlyif its rows can be partitioned into two disjoint parts satisfying


(i) if a column has two non-zero elements with the same sign thentheir rows are in different parts

(ii) if a column has two non-zero elements with opposite signs thentheir rows are in the same part

5b Integral solutions of LP problems Hoffman and Kruskal (1956)proved the following result If M is a totally unimodular matrix andb is an integral vector then for each objective function c the linearprogramming problem (LP)

maximise cx subject to Mx lt b

has an optimal solution which is integral provided that there is a finitesolution

Several optimization problems on graphs have LP formulations inwhich M is the incidence matrix or a modified form of it Amongthem are the maximum flow problem and the shortest path problemthe details of which are given in the standard text of Grotschel Lovaszand Schrijver (1988) Hoffman and Kruskals theorem leads to integral-ity results such as the fact that if the capacities are integral then thereis a maximum flow which is also integral

5c The unoriented incidence matrix As in Chapter 3 let X denotethe matrix obtained from the incidence matrix D of F by replacing eachentry plusmn1 by +1 It follows from the result of Heller and Tompkinsquoted in 5a that F is bipartite if and only if X is totally unimodularThis was first observed by Egervary (1931)

5d The image of D again With the notation of 4g if a is integer-valuedand S(w) = 0 then there is an integer-valued pound such that poundgt(pound) = w

5e The inverse of Dx Let T be a spanning tree for F and let TgtTdenote the corresponding (n mdash 1) x (n mdash 1) matrix Then (D^1)^ = plusmn1if the edge ej occurs in the unique path in T joining Vj to vn Otherwise( D r = 0

5f The Laplacian formulation of network equations For simplicity con-sider the case of a network in which each edge has conductance 1 Thenthe network equations are

z = w + n Dw = 0 Cz = 0

The last equation says that z is orthogonal to the cycle-subspace andso by Definition 46 it belongs to the cut-subspace It follows from 4a


that z = Dltgt for some potential ltfgt in the vertex-space Using the othertwo equations we obtain

D D V = Dn that is Qcjgt = n

where Q is the Laplacian matrix and 77 is a vector in which t)v is thecurrent flowing into the network at the vertex v In particular defining

+1 iiv = x- 1 Hv = y0 otherwise

we see that the solution of the network equations when a current enters at x and leaves at y is given by finding the potential satisfyingQ0 = Irfv

5g Existence and uniqueness of the solution Thomsons principle Sim-ple proofs of the results in the following paragraphs may be be found ina paper by Thomassen (1990) If x and y are vertices of a finite graphthen there is a unique solution ltjgt to the network equations for the casewhen a positive real-valued current I enters at x and leaves at y Thecurrent vector z = Dltgt is the vector satisfying Dz = Irfy for which thepower ||z||2 is a minimum (This is known as Thomsons principle)

5h An explicit solution for the network equations Suppose that x andy are adjacent vertices of a connected graph F and let K denote thetotal number of spanning trees of F (See Chapter 6 for more about K)For each spanning tree T of F send a current IK along the unique pathin T from x to y Then the current vector z which solves the networkequations for a current I entering at x and leaving at y is the sum of thesecurrents taken over all T This result goes back to Kirchhoff (1847) Forhistorical details and an algebraic proof see Nerode and Shank (1961)

5i The effective resistance For any two vertices x and y let ltfgt be thepotential satisfying Qltgt = Irfv Following Ohms law the effectiveresistance from x to y is defined to be (4gtx mdash 4gty)I- If x and y areadjacent vertices this is equal to KXVK where Kxy is the number ofspanning trees which contain the edge xy

For example it can be shown (see p 39) that the number of spanningtrees of the complete graph Kn is nn~2 since each one contains n mdash 1of the n(n mdash l)2 edges there are 2n~3 spanning trees containing agiven edge It follows that the effective resistance across an edge of Kn


is 2n In general if a graph has n vertices and m edges and it is edge-transitive (see Chapter 15) then the effective resistance across an edgeis (n - l)m

5j Monotonicity results Let R(x y T) denote the effective resistanceof F from x to y If T is obtained from F by removing an edge (thecutting operation) then

R(xyV) gt R(xyT)

The inequality is reversed if I is obtained from T by identifying twovertices (the shorting operation) These results are known as Rayleighsmonotonicity law

6

The tree-number

Several famous results in algebraic graph theory including one of theoldest are formulae for the numbers of spanning trees of certain graphsMany formulae of this kind were given in the monograph written byMoon (1970) We shall show how such results can be derived from theLaplacian matrix Q introduced in Chapter 4

Definition 61 The number of spanning trees of a graph F is itstree-number denoted by n(T)

Of course if F is disconnected then K(T) = 0 For the connectedcase Theorem 63 below is a version of a formula for K(F) which hasbeen discovered many times We need a preparatory lemma concerningthe matrix of cofactors (adjugate) of Q

Lemma 62 Let D be the incidence matrix of a graph T and letQ = DD be the Laplacian matrix Then the adjugate of Q is a multipleofJ

Proof Let n be the number of vertices of F If F is disconnected then

rank (Q) = rank (D) lt n - 1

and so every cofactor of Q is zero That is adj Q = 0 = 0JIf F is connected then the ranks of D and Q are n mdash 1 Since

Q adj Q = (det Q)I = 0

it follows that each column of adj Q belongs to the kernel of Q But thiskernel is a one-dimensional space spanned by u = [11 1] Thus

The tree-number 39

each column of adj Q is a multiple of u Since Q is symmetric so is adjQ and all the multipliers must be equal Hence adj Q is a multiple ofJ bull

Theorem 63 Every cofactor of Q is equal to the tree-number of Fthat is

adj Q = K(F)J

Proof By Lemma 62 it is sufficient to show that one cofactor of Q isequal to laquo(F) Let DQ denote the matrix obtained from D by removingthe last row then det D 0 DQ is a cofactor of Q This determinant canbe expanded by the Binet-Cauchy theorem (see Theory of Matrices byP Lancaster (Academic Press) 1969 p 38) The expansion is

det(D[)det(Dpound)

where Df denotes the square submatrix of Do whose n mdash 1 columnscorrespond to the edges in a subset U of poundT Now by Proposition 54det D[ is non-zero if and only if the subgraph (U) is a spanning tree forF and then detDy takes the values plusmn1 Since detDfy = detDy wehave det(DoDo) = K(F) and the result follows bull

For the complete graph Kn we have Q = n l - J A simple determinantmanipulation on nl mdash J with one row and column removed shows thatK(Kn) = nn~2 This result was first obtained for small values of n byCayley (1889)

We can dispense with the rather arbitrary procedure of removing onerow and column from Q by means of the following result

Proposition 64 (Temperley 1964) The tree-number of a graph Fwith n vertices is given by the formula

6(D = n-2det (J + Q)

Proof Since nJ = J2 and JQ = Owe have the following equation

(nl - J)(J + Q) = nJ + nQ - J2 - JQ = nQ

Thus taking adjugates and using Theorem 63 we can argue as followswhere K = K(F)

adj (J + Q)adj (nl - J) = adj nQ

adj (J + Q)nn~2J - nn-Jadj Q

adj (J + Q)J = nlaquoJ

(J + Q) adj (J + Q)J = (J + Q)nlaquoJ


det (J + Q)J = n2Ki

It follows that det(J + Q) = n2K as required D

The next result uses the Laplacian spectrum introduced in 4e

Corollary 65 Let 0 lt xi lt lt xn-i be the Laplacian spectrum ofa graph F with n vertices Then

IfT is connected and k-regular and its spectrum is

mi ms-i

then

laquo(T) = n-fiik - r)mr = n-V(r k)

r=l

where x denotes the derivative of the characteristic polynomial -

Proof Since Q and J commute the eigenvalues of J + Q are thesums of corresponding eigenvalues of J and Q The eigenvalues of J aren 0 0 0 so the eigenvalues of J+Q are n pi fj nn-i- Since thedeterminant is the product of the eigenvalues the first formula follows

In the case of a regular graph of degree k an (ordinary) eigenvalueA is k mdash fi where x is a Laplacian eigenvalue The result follows bycollecting the eigenvalues according to their multiplicities and recallingthat k mdash A is a simple factor of m the connected case bull

Later in this book when we have developed techniques for calculatingthe spectra of highly regular graphs we shall be able to use this Corollaryto write down the tree-numbers of many well-known families of graphsFor the moment we shall consider applications of Corollary 65 in somesimple but important cases If F is a regular graph of degree k thenthe characteristic polynomial of its line graph L(T) is known in termsof that of F (Theorem 38) If F has n vertices and m edges so that2m = nk then we have

Differentiating the result of Theorem 38 and putting A = 2k mdash 2 we get

x(L(r)2k-2) = (2k)m-nx(Tk)

Hence we obtain the tree-number of F in terms of that of L(T)

The tree-number 41

For example the tree-number of the triangle graph At = L(Kt) is

The complete multipartite graph KaXta2aa has a vertex-set which ispartitioned into s parts A A2 As where At = Oj (1 lt i lt s) twovertices axe joined by an edge if and only if they belong to different partsIn general this graph is not regular but its complement (as defined in3a) consists of regular connected components The tree-number of suchgraphs can be found by a modification of Proposition 64 due to Moon(1967) This is based on the properties of the characteristic function ofthe Laplacian matrix

ltr(r i )=det( i I-Q)

Proposition 66 (1) IfT is disconnected then the a function for Fis the product of the a functions for the components ofT

(2) IfT is a k-regular graph then a(T ) = (mdashl)nx(F k - fi) whereX is the characteristic polynomial of the adjacency matrix

(3) IfTc is the complement ofT and F has n vertices then

K(T) =n-2aTcn)

Proof (1) This follows directly from the definition of a(2) In the fc-regular case we have

det(il - Q) = det(ltI - (fcl - A)) = (-1) det((fc - n)I - A)

whence the result(3) Let Qc denote the Laplacian matrix for Fc so that Q+Qc = nlmdashJ

Then using Proposition 64 we have

K(D = n2det(J + Q) = n~2det(nl - Qc) = TC2ltT(YC n)

bull

Consider the complete multipartite graph faia2bullbullgtltraquogt where a +a + bull bull bull + as = n the complement of which consists of s compo-nents isomorphic with Kai Ka2 Kas We know that x(Knty =

(A -I- l ) n - 1 (A mdash n + 1) and using part (2) of Proposition 66 we obtain

aKa- n) = (-l)ax(Ka a - 1 - fj) = - a)01

Consequently applying parts (1) and (3) of Proposition 66

K(Kaua2a) = n-2(n)(n - a^11 (n)(n - CL)0-1

= n ~2 (n - ai)ai-1 (n - a

This result was originally found (by different means) by Austin (1960)We note the special cases

KKab) = ab-lba- K(HS) = 2 2 s -V- 1 ( s - l) s


Additional Results

6a A bound for the tree-number of a regular graph If F is a connectedfc-regular graph with n vertices then applying the arithmetic-geometricmean inequality to the product formula in Corollary 65 we obtain

lt --t I N nmdash1

1 I nkn n mdash 1

with equality if and only if F = Kn

6b More bounds for the tree-number Grimmett (1976) showed thatthe bound in 6a can be extended to non-regular graphs The result forany graph with m edges is

n - l

n n mdash 1This is clearly a generalisation of result 6a since 2m = nk in the fc-regular case Grone and Merris (1988) showed that if TT(F) is the productof the vertex-degrees then


6c A recursion for the tree-number For any (general) graph F and anyedge e which is not a loop we define the graph F ^ to be the subgraphobtained by removing e and F(e) to be the graph obtained from Fê) byidentifying the vertices of e Note that even if F itself is a graph (ratherthan a general graph) this process may produce a general graph Wehave

6d Tree-number of a Mbbius ladder The tree-number of the Mobiusladder Mh denned in 3e may be computed in two ways Using thespectral formula 65 we obtain

1 2h~1

An alternative is to use 6c to obtain a recursion formula Sedlacek(1970) used this method to obtain

nMh) = ~[(2 + v3) + (2 - v3)h] + h

The recursive method was discussed in greater generality by BiggsDamerell and Sands (1972) see 9i

The tree-number 43

6e Almost-complete graphs Let F be a graph constructed by removingq disjoint edges from Kn where n gt 2q Then

In particular taking n = 2q we have the formula for the tree-numberof if

6f Tree-numbers of planar duals Let F and F be dual planar graphs(as defined in 4f) and let D and D be the corresponding incidencematrices Suppose that F has n vertices F has n vertices and ET =ET = m then (n - 1) + (n - 1) = m If Dy is a square submatrix ofD whose n mdash 1 columns correspond to the edges of a subset U of ETand U denotes the complementary subset of ET = poundT then D[ isnon-singular if and only if D^ is non-singular Consequently

6g The octahedron and the cube The octahedron graph is H3 = -^222it is planar and the cube graph Qs is its dual We have

0 - 2 o _ 3 1 - 13 2 ] SPlaquolaquoraquo=( i 3 3

Hence K(H3) = K(QS) = 384 in agreement with 6e

6h The a function of the complement Prom the equation Q + Qc =nl mdash J we obtain

fil - Qc = [(n - M J - J - l][(n - M)I - Q]Taking determinants we have

(n - M)ltr(FcM) = (-1)

6i Spectral characterization of complete multipartite graphs The com-plete multipartite graphs defined on page 41 are the only connectedgraphs for which the second largest eigenvalue Ai is not positive (Smith1970)

Determinant expansions

In this chapter we shall investigate the characteristic polynomial xgt andthe polynomial a introduced in Chapter 6 by means of determinant ex-pansions We begin by considering the determinant of the adjacency ma-trix A of a graph F We suppose as before that VT = v v vnand that the rows and columns of A are labelled to conform with thisnotation The expansion which is useful here is the usual definition of adeterminant if A = (a^) then

det A = ] P sgn(7r)ai7rla27r2 bull bull bull antrade

where the summation is over all permutations TT of the set 1 2 nIn order to express the quantities which appear in the above expansion

in graph-theoretical terms it is helpful to introduce a new definition

Definition 71 An elementary graph is a simple graph each compo-nent of which is regular and has degree 1 or 2 In other words eachcomponent is a single edge (K2) or a cycle (C r) A spanning elementarysubgraph of F is an elementary subgraph which contains all vertices of

rWe observe that the co-rank of an elementary graph is just the numberof its components which are cycles

Proposition 72 (Harary 1962) Let A be the adjacency matrix of agraph F Then

Determinant expansions 45

where the summation is over all spanning elementary subgraphs A oF

Proof Consider a term sgn(7r)aii7ria2T2 bull bull bull olaquo7rn in the expansion ofdet A This term vanishes if for some i euro 12 n atrade = 0 thatis if vivni is not an edge of F In particular the term vanishes if nfixes any symbol Thus if the term corresponding to a permutation n isnon-zero then 7r can be expressed uniquely as the composition of disjointcycles of length at least two Each cycle (ij) of length two correspondsto the factors aâji and signifies a single edge viVj in F Eachcycle (pqr t) of length greater than two corresponds to the factorsapqaqr atp and signifies a cycle vp vqvt in F Consequentlyeach non-vanishing term in the determinant expansion gives rise to anelementary subgraph A of F with VA = VF

The sign of a permutation n is (mdashl)N where Ne is the number ofeven cycles in IT If there are cj cycles of length I then the equationEc = n shows that the number No of odd cycles is congruent to nmodulo 2 Hence

r(A) =n~(No + Ne) == Ne (mod 2)

so the sign of n is equal to (mdashl)r(A)Each elementary subgraph A with n vertices gives rise to several per-

mutations n for which the corresponding term in the determinant ex-pansion does not vanish The number of such TT arising from a given A is2SÂ since for each cycle-component in A there are two ways of choosingthe corresponding cycle in 7r Thus each A contributes (mdashl)r(A)23^ tothe determinant and we have the result bull

For example in the complete graph Kplusmn there are just two kinds of ele-mentary subgraph with four vertices pairs of disjoint edges (for whichr mdash 2 and s = 0) and 4-cycles (for which r = 3 and s = 1 There arethree subgraphs of each kind so we have

det A(K4) = 3(-l)22deg + 3(-l)321 = - 3

At the beginning of this book we obtained a description of the firstfew coefficients of the characteristic polynomial of F in terms of somesmall subgraphs of F (Proposition 23) We shall now extend that resultto all the coefficients We shall suppose as before that

X(F A) - Xn + cxA1 + c2A-2 + + cn

Proposition 73 The coefficients of the characteristic polynomial aregiven by


where the summation is over all elementary subgraphs A of F with ivertices

Proof The number -)lCi is the sum of all principal minors of Awith i rows and columns Each such minor is the determinant of theadjacency matrix of an induced subgraph of F with i vertices Anyelementary subgraph with i vertices is contained in precisely one of theseinduced subgraphs and so by applying Proposition 72 to each minorwe obtain the required result bull

The only elementary graphs with fewer than four vertices are K2 (anedge) and C3 (a triangle) Thus we can immediately regain the resultsof Proposition 23 from the general formula of Proposition 73 Wecan also use Proposition 73 to derive explicit expressions for the othercoefficients for example c Since the only elementary graphs with fourvertices are the cycle graph C4 and the graph having two disjoint edgesit follows that

C4 = na - 2rib

where na is the number of pairs of disjoint edges in F and rib is thenumber of 4-cycles in F (See 7i)

As well as giving explicit expressions for the coefficients of the charac-teristic polynomial Proposition 73 throws some light on the problem ofcospectral graphs (2f) The fact that elementary subgraphs are ratherloosely related to the structure of a graph helps to explain why there aremany pairs of non-isomorphic graphs having the same spectrum Thisis particularly so in the case of trees (see 7b and 7c)

We now turn to an expansion of the characteristic function of theLaplacian matrix

ltx(F H) = detOil - Q)

Although the Laplacian matrix Q differs from mdashA only in its diagonalentries the ideas involved in this expansion are quite different from thosewhich we have used to investigate the characteristic polynomial of AOne reason for this is that a principal submatrix of Q is (in general) notthe Laplacian matrix of an induced subgraph of F (the diagonal entriesgive the degrees in F rather than in the subgraph)

We shall write

CT(F H) = det(tl - Q) = nn + q-ii1 + + laquo-iM + qnThe coefficient (mdash1)^ is the sum of the principal minors of Q whichhave i rows and columns Using results from Chapter 6 and some simple


observations we obtain

qi = -2ET gn_x = (-ly^ncOT) gn = 0We shall find a general expression for qi which subsumes these resultsThe method is based on the expansion of a principal minor of Q = DDby means of the Binet-Cauchy theorem as in the proof of Theorem 63

Let X be a non-empty subset of the vertex-set of F and Y a non-emptysubset of the edge-set of F We denote by D(X Y) the submatrix of theincidence matrix D of F defined by the rows corresponding to vertices inX and the columns corresponding to edges in Y The following lemmaamplifies the results of Propositions 53 and 54

Lemma 74 Let X and Y be as above with X = Y and let VQdenote the vertex-set of the subgraph (Y) Then D(XY) is invertible ifand only if the following conditions are satisfied

(1) X is a subset ofVo(2) (Y) contains no cycles(3) VQX contains precisely one vertex from each component of (Y)

Proof Suppose that D(XY) is invertible If X were not a subsetof VQ then T)(XY) would contain a row of zeros and would not beinvertible hence condition (1) holds The matrix ~D(VoY) is the inci-dence matrix of (Y) and if (Y) contains a cycle then D(Vo Y)z = 0 forthe vector z representing this cycle Consequently D(X Y)z = 0 andTgt(XY) is not invertible Thus condition (2) holds It follows that theco-rank of (Y) is zero that is

where c is the number of components of (Y) Since X = Y we have]TioX| = c If X contained all the vertices from some component of (V)then the corresponding rows of D(X Y) would sum to 0 and D(X Y)would not be invertible Thus VQX contains some vertices from eachcomponent of (Y) and since VQ X = c it must contain precisely onevertex from each component and condition (3) is verified

The converse is proved by reversing the argument bull

A graph $ whose co-rank is zero is a forest it is the union of compo-nents each of which is a tree We shall use the symbol p(ltpound) to denote theproduct of the numbers of vertices in the components of $ In particularif $ is connected it is a tree and we have

Theorem 75 The coefficients qi of the polynomial ltr(F n) are givenby the formula


where the summation is over all sub-forests $ oF which have i edges

Proof Let Qx denote the principal submatrix of Q whose rows andcolumns correspond to the vertices in a subset X of VF Then ltjj =J^detQx where the summation is over all X with X = i Using thenotation of Lemma 74 and the fact that Q mdash DD it follows from theBinet-Cauchy theorem that

)2det Qx = ^2 det T)(X Y) det DX Yf = ^ ( d e t D(X Y))2

This summation is over all subsets Y of ET with Y = X = i Thus

XY

By Proposition 53 (detD(X Y))2 is either 0 or 1 and it takes thevalue 1 if and only if the three conditions of Lemma 74 hold For eachforest $ = (Y) there are p($) ways of omitting one vertex from eachcomponent of $ and consequently there are ygt($) summands equal to 1in the expression for qi This is the result

Corollary 76 The tree-number of a graph T is given by the formula

where the summation is over all forests $ which are subgraphs of thecomplement ofT

Proof The result of Proposition 66 part (3) expresses K(T) in termsof the a function of Fc The stated result follows from the formula ofTheorem 75 for the coefficients of a bull

This formula can be useful when the complement of F is relativelysmall examples of this situation are given in 6e and 7d In the case of aregular graph F the relationship between a and x leads to an interestingconsequence of Theorem 75

Proposition 77 LetT be a regular graph of degree k and let^ (0 lti lt n) denote the ith derivative of the characteristic polynomial of FThen

where the summation is over all forests $ which are subgraphs of F withEamp = n-iProof Prom part (2) of Proposition 66 we have


The Taylor expansion of at the value k can be written in the form

i=0Comparing this with a(T (i) = J2 Qn-iJ-1 we have the result bull

We notice that the case i = 1 of Proposition 77 gives

which is just the formula given in Corollary 65

Additional Results

7a Odd cycles (Sachs 1964) Let x(f A) = Yl Cn-il and suppose

C3 = C5 = bull bull bull = C2r-1 = 0 C2r+1 0

Then the shortest odd cycle in F has length 2r + 1 and there aremdashC2r+i2 such cycles

7b The characteristic polynomial of a tree Suppose that J^ CjAtrade~1 isthe characteristic polynomial of a tree with n vertices Then the oddcoefficients c-zr+i are zero and the even coefficients cltiT are given by therule that (mdashl)rC2r is the number of ways of choosing r disjoint edges inthe tree

7c Cospectral trees The result 7b facilitates the construction of pairsof cospectral trees For example there are two different trees with eightvertices and characteristic polynomial A8 mdash 7A6 + 10A4 Schwenk (1973)proved that if we select a tree T with n vertices all such trees beingequally likely then the probability that T belongs to a cospectral pairtends to 1 as n tends to infinity

7d The a function of a star graph A star graph is a complete bipartitegraph Kifi For such a graph we can calculate a explicitly from theformula of Theorem 75 the result is

Consequently if F is the graph obtained by removing a star K$ fromKn where n gt b + 1 we have


7e Complete matchings Hamiltonian cycles and the determinant for-mula We may write the formula for det A as

where f(r s) is the number of spanning elementary subgraphs with rankr and co-rank s Two terms in this formula have special significanceThe number finji 0) is the number of disjoint edges which cover allthe vertices - the complete matchings The number f(n mdash 11) is thenumber of spanning elementary subgraphs which are connected that isthe number of single cycles which cover all the vertices - the Hamiltoniancycles

7f Reconstruction - Kellys lemma For each vertex v 6 VF let Fv

denote the induced subgraph (VT v) The deck of F is the set of(unlabelled) induced subgraphs Tv | v S VT The graph is said to bereconstructible if every graph with the same deck as F is isomorphic toF The reconstruction conjecture is that every graph with at least threevertices is reconstructible

A function defined on graphs is said to be reconstructible if it takesthe same value on all graphs with the same deck For any graphs F andA let n(F A) be the number of subgraphs of F which are isomorphic toA Standard double counting arguments lead to the formula

(rA) pound ( r A )

From this formula it follows that n(F A) is reconstructible whenever|VA| lt |VT| (Kelly 1957)

7g Reconstruction - Kocays Lemma A sequence of graphs

is said to be a cover of the graph F if there are subgraphs A of F suchthat A is isomorphic to ltfgti (1 lt i lt I) and the union of the subgraphsis F The number of covers of F by T is denoted by c(F^r) Kocay(1981) proved that provided all the members of the sequence T havefewer vertices than F the function

xis reconstructible where the sum is taken over all isomorphism classesof graphs X such that VX = |VT|


7h The reconstructibility of the characteristic polynomial Using thelemmas of Kelly and Kocay and the formula in Proposition 73 it canbe shown that the coefficients c of the characteristic polynomial arereconstructible In particular (mdashl)nCn mdash det A is reconstructible Theseresults were first established by Tutte (1979) using a different methodHis proof and that using Kocays lemma (as given by Bondy (1991))both depend on showing that the number of Hamiltonian cycles denotedby f(n - 11) in 7e is reconstructible

7i Angles and the number of 4-cycles The number of 3-cycles in a graphis determined by the spectrum (see 2h) but the number of 4-cycles isnot except in special cases such as when the graph is regular Howeverthe number of 4-cycles is determined by the spectrum and the angleswhich are defined as follows Let ]T] AaEo be the spectral decompositionof the adjacency matrix A as denned in 2j and let ei e2 bull en be thestandard orthonormal basis for Euclidean n-space Then the angles arethe numbers

aij=Eiej (lltijltn)

More about this construction and an explicit formula for the number of4-cycles can be found in a paper by Cvetkovic and Rowlinson (1988)

7j The Shannon capacity of a graph Let FA denote the product ofgraphs F and A obtained by taking the vertex-set to be Cartesian prod-uct of their vertex-sets and defining two distinct vertices to be adjacentif both coordinates are equal or adjacent Let F r denote the productof r copies of F and let a(Fr) denote the maximum number of mutu-ally non-adjacent vertices of F r A construction in coding theory due toShannon involves the quantity

8(F) = l im(a(F r))1 r rmdashgtoo

and this is known as the Shannon capacity of F Since a(F) r lt a(F r) itfollows that a(T) lt 9(F) but in general equality does not hold Lovasz(1979) showed that O(F) is bounded above by the largest eigenvalue ofany real symmetric matrix C for which c^ = 1 whenever Vi and Vj arenot adjacent In particular this yields the result 0(Cs) = Vo

8

Vertex-partitions and the spectrum

One of the oldest problems in graph theory is the vertex-colouring prob-lem which involves the assignment of colours to the vertices in sucha way that adjacent vertices have different colours This can be inter-preted as a problem about a special kind of partition of the vertex-setas described in the first definition below In this chapter we shall applyspectral techniques to the vertex-colouring problem using inequalitiesinvolving the eigenvalues of a graph Similar techniques can also be ap-plied to other problems about vertex-partitions and some of these arementioned in the Additional Results at the end of the chapter

Definition 81 A colour-partition of a general graph F is a partitionof VT into subsets called colour-classes

vr = v1uv2uuvlsuch that each Vj (1 lt i lt I) contains no pair of adjacent vertices Inother words the induced subgraphs (Vi) have no edges The chromaticnumber of F written f(F) is the least natural number I for which sucha partition is possible

We define a vertex-colouring of F to be an assignment of colours to thevertices with the property that adjacent vertices have different coloursso clearly a vertex-colouring in which I colours are used gives rise to acolour-partition with I colour-classes

We note that if F has a loop then it has a self-adjacent vertex andconsequently no colour-partitions Also if F has several edges joiningthe same pair of vertices then only one of these edges is relevant to

Vertex-partitions and the spectrum 53

the definition of a colour-partition since the definition depends only onwhether vertices are adjacent or not Thus we can continue for themoment to deal with strict graphs However this is allowable only forthe purposes of the present chapter some of the constructions used inPart Two require the introduction of general graphs

If i(r) = 1 then F has no edges If vT) mdash 2 then F is a bipartitegraph as denned in 2c Since a cycle of odd length cannot be colouredwith two colours it follows that a bipartite graph contains no odd cyclesThis observation leads to another proof of the result established in 2c

Proposition 82 Suppose the bipartite graph T has an eigenvalue Aof multiplicity m(A) Then mdash A is also an eigenvalue ofT and m(mdashX) =m(A)

Proof The formula of Proposition 73 expresses the characteristicpolynomial of a graph F in terms of the elementary subgraphs of FIf F is bipartite then F has no odd cycles and consequently no ele-mentary subgraphs with an odd number of vertices It follows that thecharacteristic polynomial of F has the form

X(F z) = zn + c2zn~2 + CiZ1- + = z6pz2)

where 6 = 0 or 1 and p is a polynomial function Thus the eigenvalueswhich are the zeros of have the required property bull

The spectrum of the complete bipartite graph Kaltb can be found inthe following manner We may suppose that the vertices of Ka^ arelabelled in such a way that its adjacency matrix is

A - f deg JA ~ [J 0

where J is the a x b matrix having all entries +1 The matrix A hasjust two linearly independent rows and so its rank is 2 Consequently0 is an eigenvalue of A with multiplicity a + b mdash 2 The characteristicpolynomial is thus of the form za+b~2(z2 +C2) By Proposition 23 mdash cîs equal the number of edges of Ka^ that is 06 Hence

This example illustrates the fact (Proposition 82) that the spectrumof a bipartite graph is symmetrical with respect to the origin Indeedthe converse of this result is also true [CvDS p 87] But if i(r) gt 2the spectrum of F does not have a distinctive property as it does inthe bipartite case However as we shall see it is possible to makeuseful deductions about the chromatic number from a knowledge of themaximum and minimum eigenvalues of F


For any real symmetric matrix M we shall denote the maximum andminimum eigenvalues of M by Amax(M) and Amin(M) If M is theadjacency matrix of a graph F we shall also use the notation Amax(F)and Amn(F) It follows from Proposition 82 that for a bipartite graphF we have Amin(r) = -A m a x (F)

We need a useful technique from matrix theory Let (x y) denote theinner product of the column vectors x y For any real n x n symmetricmatrix X and any real non-zero n x l column vector z the number(z Xz)(z z) is known as the Rayleigh quotient and written i(Xz)In matrix theory it is proved that

Am x(X)gt JR(X)gtA I I l l l l(X) for all z jk 0

a result which has important applications in spectral graph theory

Proposition 83 (1) If A is an induced subgraph ofT then

Amax(A) lt A r a a x(F) Amin(A) gt A r a i n(F)

(2) If the greatest and least degrees among the vertices ofT are fcmax(F)and fcmjn(F) and the average degree is fcaVe(F) then

) gt Amax(F) gt Awe(r) gt fcmin(F)

Proof (1) We may suppose that the vertices of F are labelled sothat the adjacency matrix A of F has a leading principal submatrixAo which is the adjacency matrix of A Let Zo be chosen such thatAozo = Amax(Ao)zo and (zoZo) = 1- Further let z be the columnvector with |VT| rows formed by adjoining zero entries to Zo- Then

Amax(A0) = -R(A0z0) = R(Az) lt Amax(A)

That is Amax(A) lt Amax(F) The other inequality is proved similarly(2) Let u be the column vector each of whose entries is + 1 Then if

n = |VT| and fcW is the degree of the vertex Vi we have

R(A u) = plusmn pound ay = i pound laquo = fcave(F)ij i

The Rayleigh quotient i(A u) is at most Amax(A) that is Amax(F) andit is clear that the average degree is not less than the minimum degreeHence

Amax(F) gt fcave(F) gt kmin(T)

Finally let x be an eigenvector corresponding to the eigenvalue Ao =Amax(F) and let Xj be a largest positive entry of x By an argumentsimilar to that used in Proposition 31 we have

= (Aox)j = Exi lt k^Xj lt kmax(r)Xj


where the sum E is taken over the vertices Vi adjacent to Vj Thus() n

We shall now bound the chromatic number of F in terms of Amax(r)and Amin(r) A graph F is l-critical if i(F) = and for all inducedsubgraphs A ^ F w e have u(A) lt I

Lemma 84 Suppose F is a graph with chromatic number I gt 2 ThenF has an l-critical induced subgraph A and every vertex of A has degreeat least I - 1 in A

Proof The set of all induced subgraphs of F is non-empty and containssome graphs (for example F itself) whose chromatic number is I andalso some graphs (for example those with one vertex) whose chromaticnumber is not Let A be an induced subgraph whose chromatic numberis I and which is minimal with respect to the number of vertices thenclearly A is Z-critical If v is any vertex of A then VA v) is an inducedsubgraph of A and has a vertex-colouring with l mdash l colours If the degreeof v in A were less than l mdash l then we could extend this vertex-colouringto A contradicting the fact that v(A) = I Thus the degree of v is atleast I - 1 bull

Proposition 85 (Wilf 1967) For any graph T we have

uT) lt 1 + Amax(r)

Proof It follows from Lemma 84 that there is an induced subgraphA of T such that v(A) = u(T) and kmin(A) gt i(T) - 1 Thus using theinequalities of Proposition 83 we have

Kr ) lt i + fcmin(A) lt i + Amax(A) lt i + Amax(r)

bull

Wilfs bound may be compared with the simple bound u lt 1 + fcmaxwhich is proved by an obvious argument There is also a nontrivialrefinement of the simple bound known as Brookss theorem v lt kmaxunless F is a complete graph or an odd cycle For example for thecomplete bipartite graph Ka$ we have

kmaxKab) = max(a b) Xmax(Katb) = y(ab)

When a is large in comparison with 6 the second number is much smallerthan the first but it is still a poor bound for the chromatic number 2

Our next major result is complementary to the previous one in thatit provides a lower bound for the chromatic number We require a pre-liminary lemma and a corollary


Lemma 86 Let X be a real symmetric matrix partitioned in the form

P QlQ R j

where P and R are square symmetric matrices Then

Proof Let A = Amjn(X) and take an arbitrary e gt 0 Then X =X mdash (A mdash e)I is a positive-definite symmetric matrix partitioned in thesame way as X with

P = P - (A - e)I Q = Q R = R - (A - e)IBy applying the method of Rayleigh quotients to the matrix X it canbe shown that

Amax(X ) lt Am a x( ) + Amax(rL )

(See for instance Linear Transformations by H L Hamburger and M EGrimshaw (Cambridge 1956) p 77) Thus in terms of X P and Rwe have

Amax(X) - (A - e) lt Amax(P) - (A - e) + Amax(R) - (A - c)

and since c is arbitrary and A = Amin(X) we have the result bull

Corollary 87 Let A be a real symmetric matrix partitioned into t2

submatrices Aj in such a way that the row and column partitions arethe same in other words each diagonal sub-matrix An (1 lt i lt t) issquare Then

t

- l)Amin(A) lt VA m a x (Ai i )

Proof We prove this result by induction on t It is true when t = 2 bythe lemma Suppose that it is true when t = T mdash 1 then we shall showthat it holds when t = T Let A be partitioned into T2 submatricesin the manner stated and let B be the matrix A with the last row andcolumn of submatrices deleted By the lemma

Amax(A) + Amin(A) lt Amax(B) + Amax(ATT)

and by the induction hypothesisT-l

Amax(B) + (T - 2)Amin(B) lt ] T Amax(A i i)

Now Amin(B) gt Amjn(A) as in the proof of Proposition 83 Thusadding the two inequalities we have the result for t = T and the generalresult follows by induction bull

We can now establish a lower bound for the chromatic number


Theorem 88 (Hoffman 1970) For any graph F whose edge-set isnon-empty

Proof The vertex-set VF can be partitioned into v = i(F) colour-classes consequently the adjacency matrix A of F can be partitionedinto v2 submatrices as in the preceding corollary In this case thediagonal submatrices An (1 lt i lt u) consist entirely of zeros and soAmax(Aj) = 0 (1 lt i lt v) Applying Corollary 87 we have

Amax(A) + (i - l)Amin(A) lt 0

But if F has at least one edge then Amin(A) = Amjn(r) lt 0 The resultnow follows bull

In cases where the spectrum of a graph is known Hoffmans boundcan be very useful Consider for example the graph E which arisesfrom the classical configuration of 27 lines on a general cubic surface inwhich each line meets 10 other lines The vertices of pound represent linesand adjacent vertices represent skew lines so that pound is a regular graphwith 27 vertices and degree 16 This is the graph with least eigenvaluemdash2 mentioned in 3g Since Amax(pound) = 16 and Amjn(pound) = -2 Hoffmansbound is v(E) gt 1 + 162 = 9 a result which would be difficult toestablish by direct means On the other hand it is fairly easy to finda vertex-colouring using 9 colours (Haemers 1979) so Hoffmans boundleads to the exact answer v(pound) = 9 in this case

Additional Results

8a The eigenvalues of a planar graph Let F be a planar connectedgraph Then it follows from Theorem 88 and the four-colour theoremthat

Amin(A ) S 7jAmax(l )bull

8b Another bound for the chromatic number Let F be a regular graphof degree k with n vertices In any colour-partition of F each colour-classhas at most n mdash k vertices consequently i(F) gt n(n mdash k) Cvetkovic(1972 see also 8h) proved a corresponding result for any not necessarilyregular graph

ltrgt s


8c The second eigenvalue of the Laplacian The eigenvalues of a realsymmetric matrix may be characterised in terms of the Rayleigh quo-tient In particular for the first non-zero eigenvalue MI of the Laplacianmatrix Q the characterisation asserts that

Mi = min i(Qx)ux=O

where u is the all-1 vector the eigenvector corresponding to Mo- Thisprovides a powerful method for finding upper bounds for Mi- If we thinkof x as a real-valued function pound defined on the vertex-set the condi-tion ux = 0 becomes J2 euro(v) mdash 0 anlt3 for any function satisfying thiscondition we have (by a simple manipulation of the Rayleigh quotient)

vweE v

8d A spectral bound for the isoperimeiric number Let 8X be the cutdefined by X C VT and let x = X n = |V17| Define f (v) to be Ia ifv pound X and mdash l(n mdash x) otherwise Then 8c implies that

Mi lt || (-x n mdash x

It follows that for the isoperimetric number defined in 4b we have(F) gt Mi2 and in the fc-regular case i(T) gt (k mdash X)2 (Alon andMilman 1985)

8e Equipartitions Suppose that T is a graph with n vertices and X isa partition of its vertex-set Let 6X denote the set of edges whose endsare in different parts We say that X is an equipartition if each part hasthe same size p then n = pq where q is the number of parts BiggsBrightwell and Tsoubelis (1992) showed that in this case

Mi lt SX ( 2

8f The odd graphs (Biggs 1979) Let k be a natural number greater than1 and let 5 be a set of cardinality 2k mdash 1 The odd graph Ok is defined asfollows its vertices correspond to the subsets of S of cardinality k mdash 1and two vertices are adjacent if and only if the corresponding subsetsare disjoint (For example O2 = -K3 and O3 is the Petersen graph) Okis a regular graph of degree k when k mdash 2 its girth is 3 when k = 3 itis 5 and when k gt 4 it is 6

The spectrum of Ok can be obtained by the methods described inChapters 20 and 21 (see 21b) In particular the largest eigenvalue Amax


is k the next largest is A mdash 2 and the least is 1 mdash k Using Theorem 88and 8d respectively we get the following lower bounds for the chromaticand isoperirnetric numbers

= 1

To see how good these bounds are let V[a] denote the set of verticescontaining a given pair a0 euro S let V[a ] denote the set of verticescontaining a but not 3 and so on Then the three sets V[a 0 Va 0and V[a 3]uV[a ] form a colour partition so v(Ok) = 3 Furthermorethe cut denned by X mdash V[a ] U V[a 0 and its complement satisfies

2(2r23) k

Thus i(Ofc) lt 1 + (k mdash I ) 1 Further results about the odd graphs maybe found in 17d 20b and 21b

8g The Motzkin-Straus formula Consider the quadratic programming

problem (QP)

maximize xAx subject to ux = 1 x gt 0

where A is the adjacency matrix of a graph F Define the support of afeasible vector x to be the set of vertices vt for which xraquo ^= 0 It canbe shown that for an optimal x with minimal support the support is aclique (a complete subgraph) in T It follows that the maximum valuefor the QP is 1 mdash lw(r) where co(F) is the size of the largest clique inF This is the formula of Motzkin and Straus (1965) Putting Xi = 1nfor i = 12n = |VT| and letting m - ET we get

1 gtw(r) - n2

In particular we have Turdns Theorem if F has no triangles thenm lt n24

8h Another spectral bound Let s be the sum of the entries of the nor-malized eigenvector corresponding to Amax Wilf (1985) observed thatthe Motzkin-Straus formula leads to the result w gt s2(s2 mdash Amax) Sinces2 lt n it follows that u gt n(n mdash Amax) Furthermore the chromaticnumber v cannot be less than w so this strengthens the result 8b ofCvetkovic

PART TWO

Colouring problems

9

The chromatic polynomial

Part Two is concerned with polynomial functions which represent certainnumbers associated with graphs The best-known example the chro-matic polynomial is introduced in this chapter It should be stressedthat here we have to deal with general graphs because some of theconstructions fail when restricted to strict graphs

Definition 91 Let F be a general graph with n vertices and let ube a complex number For each natural number r let mr(F) denotethe number of distinct colour-partitions of VT into r colour-classes anddefine laquo(r) to be the complex number u(u mdash l)(u mdash 2) (u mdash r +1) Thechromatic polynomial of F is the function defined by

Proposition 92 If s is a natural number then C(Ts) is the numberof vertex-colourings of F using at most s colours

Proof Every vertex-colouring of F in which exactly r colours are usedgives rise to a colour-partition into r colour-classes Conversely for eachcolour-partition into r colours we can assign s colours to the colour-classes in s(s mdash 1) (s mdash r + 1) ways Hence the number of vertex-colourings in which s colours are available is J2mr(F)s(r) = C(F s)

bull

64 Colouring problems

The simplest example is the chromatic polynomial of the completegraph Kn Since every vertex of Kn is adjacent to every other one thenumbers of colour-partitions are

miKn) = m2(Kn) = = mn-iKn) = 0 mn(Kn) = 1

Hence

C(Kn u) = uu - l)(w - 2) (u - n + 1)

Possibly the most important fact about the chromatic polynomial isthat it is indeed a polynomial in other words the number of vertex-colourings of a graph with a given number of colours available is thevalue of a polynomial function This is because the expressions U(r)which occur in the definition are themselves polynomials

Some simple properties of the chromatic polynomial follow directlyfrom its definition For example if F has n vertices then mn(T) = 1hence C(F u) is a monic polynomial of degree n Other results followdirectly from Proposition 92 and the principle that a polynomial isuniquely determined by its values at an infinite set of natural numbersFor instance if F is disconnected with two components Fi and F2 thenwe can colour the vertices of Fi and F2 independently and it follows thatC(Ts) = C(Fi s)C(F2 s) for any natural number s Consequently

C(r u)=C(r i u)C(r2u) as elements of the ring of polynomials with integer coefficients

Since u is a factor of U(r) for all r gt 1 it follows that C(F0) = 0for any general graph F If F has c components then the coefficients of1 = udeg u1 uc~l are all zero by virtue of the result on disconnectedgraphs in the previous paragraph Also if ET 0 then F has no vertex-colouring with just one colour and so C(F 1) = 0 and u mdash 1 is a factorof C(ru)

The problem of finding the chromatic number of a graph is part of thegeneral problem of locating the zeros of its chromatic polynomial be-cause the chromatic number ^(F) is the smallest natural number u whichis not a zero of C(F u) This fact has stimulated some interesting work(see 9i 9j and 9k for example) but as yet elementary methods haveproved more useful in answering questions about chromatic numbers

The simplest method of calculating chromatic polynomials is a recur-sive technique Suppose that F is a general graph and that e is an edgeof F which is not a loop The graph Fê whose edge-set is ET e andwhose vertex-set is VF is said to be obtained by deleting e while thegraph F(ej constructed from Fê^ by identifying the two vertices incidentwith e in F is said to be obtained by contracting e We note that F ^

The chromatic polynomial 65

has one edge fewer than F and F(ej has one edge and one vertex fewerthan F and so the following Proposition provides a method for calculat-ing the chromatic polynomial by repeated reduction to smaller graphsThis is known as the deletion-contraction method

Proposition 93 The chromatic polynomial satisfies the relationC(Tu) = C(rltegtlaquo) - C(T(e)u)

Proof Consider the vertex-colourings of F ^ with s colours availableThese colourings fall into two disjoint sets those in which the ends ofe are coloured differently and those in which the ends of e are colouredalike The first set is in bijective correspondence with the colourings ofF and the second set is in bijective correspondence with the colouringsof F(e) Hence C(T^ s) = C(F s) + C(F(e) s) for each natural numbers and the result follows bull

Corollary 94 If T is a tree with n vertices then

C(Tu)=uu-l)n-Proof We prove this by induction using the elementary fact that anytree with n gt 2 vertices has a vertex (in fact at least two vertices) ofdegree 1 The result is clearly true when n = 1 Suppose it is true whenn = N mdash 1 and let T be a tree with TV vertices e an edge of T incidentwith a vertex of degree 1 Then T^ has two components an isolatedvertex and a tree with N mdash 1 vertices the latter being T(e) Hence

and using Proposition 93 and the induction hypothesis

CTlaquo) = (u - l)C(T(e) u) = (laquo - l)u(laquo - I ) 2 = u(u - )N~X

Hence the result is true when n = N and for all n by the principle ofinduction bull

The deletion-contraction method also yields the chromatic polynomialof a cycle graph Cn If n gt 3 the deletion of any edge from Cn resultsin a path graph Pn which is a tree with n vertices and the contractionof any edge results in a cycle graph Cn-- Hence

C(Cnu) = u(u - I ) - 1 - C(Cn_ i u)Since C3 = K3 we have

C(C3 u) = uu - )u - 2) = (u - I)3 - (w - 1)We can solve the recursion given above with this initial condition toobtain the formula


We now describe two other useful techniques for calculating chromaticpolynomials The first is concerned with the join operation for graphsSuppose Fi and F2 are two graphs then we define their join Fj + F2 tobe the graph with vertex-set and edge-set given by

+ F2) = ETi U ET2 U x yxeVTuye VT2

In other words Fi + F2 consists of copies of Fi and F2 with additionaledges joining every vertex of Fi to every vertex of F2

Proposition 95 The numbers of colour-partitions of F = Ti + T2

are given by

mi(T) =

Proof Since every vertex of Fj is adjacent (in F) to every vertex ofF2 any colour-class of vertices in F is either a colour-class in Fj or acolour-class in F2 Hence the result bull

Corollary 96 The chromatic polynomial of the join Ti + F2 is

C(Ti + F2 u) = C(Tiu) o C(F2u)

where the o operation on polynomials signifies that we write each poly-nomial in the form 2ZTOiM(i) and multiply as ifu^ were the power ul

D

For example the complete bipartite graph -^33 is the join V3 + JV3where Nn is the graph with n vertices and no edges From Corollary96 we have

+ 3U(2) + U(i)) O (u(3) + 3U(2) + U

6U(5) + llU(4) + 6W(3) + W(2)

M5 + 36w4 - 75u3 + 78u2 - 31u

The chromatic polynomials of all complete multipartite graphs can befound in this way

Another application of the method yields the chromatic polynomialsof the graphs N + F and N2 + F sometimes known as the cone andsuspension of F and denoted by cT and sT respectively


Proposition 97 The chromatic polynomials of a cone and a suspen-sion are given by

C(cT u) = uC(F u - 1)

C(sT u) = uu - 1)C(F u - 2) + uC(T u - 1)

Proof Let C(T u) = ^2 miU^) Using Corollary 96 and the fact thatU(i+i) = uu - l)(j) we have

C(cT u) = C(Nx +Tu)=uo C(T u) = u(1) o

j(w - l)(j) = uC(Tu - 1)

The second part is proved in a similar way using the identity u2 mdashD

Another useful technique for the calculation of chromatic polynomialsapplies to graphs of the kind described in the next definition

Definition 98 The general graph V is quasi-separable if there is asubset K of VT such that the induced subgraph (K) is a complete graphand the induced subgraph (VT K) is disconnected T is separable if| AT | lt 1 in this case either K = 0 so that T is in fact disconnected orK = 1 in which case we say that the single vertex of K is a cut-vertex

It follows that in a quasi-separable graph V we have VT = V U V2where (Vi Pi V2) is complete and there are no edges in F joining V (Vi n V2) to V2 (V n V2) We shall refer to the pair (Vi V2) as aquasi-separation of F or simply a separation if |Vi PI V2I lt 1

A graph which is quasi-separable but not separable is shown in Figure3 the relevant quasi-separation is given by Vi = 1 24 V2 = 234

Figure 3 a quasi-separable graph

Proposition 99 If the graph F has quasi-separation (Vi V2) then

C(Fu) =


Proof If V (1 Vi is empty we make the convention that the denomina-tor is 1 and the result is a consequence of the remark about disconnectedgraphs following Proposition 92 Suppose that (Vi n V2) is a completegraph Kt t gt 1 Since F contains this complete graph F has no vertex-colouring with fewer than t colours and so u^ is a factor of C(Tu)For each natural number s gt t C(F s)s(t) is the number of ways ofextending a given vertex-colouring of Vi D V2) to the whole of F us-ing at most s colours Also both Vi and (V2) contain the completegraph Kt = (V n V2) so C(Vi s)s(t) i euro 12 has a correspondinginterpretation Since there are no edges in F joining V V D Vj) toV2 (Vi n V2) the extensions of a vertex-colouring of Vi n V2) to Vi)and to V2) are independent Hence

C(Ta)

for all s gtt The corresponding identity for the polynomials follows

bullThe formula of Proposition 99 is often useful in working out chromatic

polynomials of small graphs For instance the graph shown in Figure 3is two K3s with a common K2 hence its chromatic polynomial is

( - ) ( laquo - 2 ) laquo ( u - ) ( laquo - 2 ) = u(u _ 1 ) ( n _ 2)2_

An important theoretical application of Proposition 99 will be describedin Chapter 12

Additional Results

9a Wheels and pyramids The cone of the cycle graph Cn- is the wheelor pyramid Wn the suspension of Cn-i is the double pyramid Un Thechromatic polynomials of these graphs are

C(Wn u) = u(u - 2)71-1 -(- ( - l ) - 1 ^ - 2)

In u) = u(u - l)(u - 3)n-2 + u(u - 2)n~2 + (-l)nu(u2 - 3u + 1)

9b The cocktail-party graphs Let ps(u) mdash C(Hsu) where Hs is thecocktail-party graph 222 with 2s vertices The polynomials ps(u)can be found from the recursion

pi(u)=u2 ps(u) = u(u - I)p3-i(u - 2) + ups-i(u - 1) (s gt 2)


9c Ladders and Mobius ladders The ladder Lh (h gt 3) is a regulargraph of degree 3 with 2h vertices u v2 bull bull bull laquoh v v Vh the ver-tices u Uh form a cycle of length h as do the vertices v Vhand the remaining edges are of the form ujigt 1 lt i lt h The Mobiusladders Mh were defined in 3e By systematic use of the deletion-contraction method Biggs Damerell and Sands (1972 see also 9i)showed that

C(Lh u) = (u2 -3u + 3)h + (u- 1)(3 - u)h + (1 - u)h) + u2 -

C(Mh u) = (u2 -3u + 3)h + (u - 1)(3 - u)h - (1 - u)h - 1

9d The chromatic polynomial characterizes trees Corollary 94 impliesthat different graphs may have the same chromatic polynomial since anytwo trees with the same number of vertices have this property Howeverif F is a simple graph with n vertices and C(T u) = uu - I)trade1 thenT is a tree (Read 1968)

9e Chromatically unique graphs A graph is said to be chromaticallyunique if it is the only graph with its chromatic polynomial FromCorollary 94 we know that any tree with more than three vertices isnot chromatically unique Several families of graphs are known to bechromatically unique among them the following

(a) The complete graphs Kn(b) The cycle graphs Cn (n gt 3)(c) The wheel graphs Wn for odd n

It is known that Wsect and W$ are not chromatically unique but Wo isSee Li and Whitehead (1992) for this result and additional references

9f The chromatic polynomials of the regular polyhedra The chromaticpolynomials of the graphs formed by the vertices and edges of the fiveregular polyhedra in three dimensions are known The graph of thetetrahedron is K4 the graph of the octahedron is H3 = K222 (9b)and the graph of the cube is L4 (9c) The chromatic polynomial ofthe icosahedron was computed by Whitney (1932b) after removing thefactors u(u mdash l)(u mdash 2)(u mdash 3) it is

u8 -24u7+260u6 - 1670u5+6999w4 - 19698u3+36408u2 -40240u+20170

The computation of the chromatic polynomial of the dodecahedron wasfirst attempted by DA Sands (in an unpublished thesis 1972) andHaggard (1976) In order to reduce the size of the coefficients it is


convenient to express the result in the form17

-u(u - l)(w - 2) ] T Ci(l - u)

In this form the coefficients Ci are all positive and they are

1 10 56 230 759 2112 5104 10912 20880 35972

55768 77152 93538 96396 80572 50808 21302 4412

9g Interpolation formulae Suppose that two finite sequences of realnumbers mo m i mn and PoPi bull bull bull pn are related by the rule

k

Pk = ^ m r f c ( r ) r=0

Then there is an inverse formula giving the ms in terms of the ps andthis in turn leads to a formula for the polynomial p(u) of degree n whosevalue at k euro 01 n is pk-

2fc=0 V r = 0 fc=0 V

In particular we have formulae for the numbers of colour-partitions andthe chromatic polynomial in terms of the numbers of fc-colourings

9h Acyclic orientations An orientation of a graph as defined in Chapter4 is said to be acyclic if it has no directed cycles For example on atree with n vertices any orientation is acyclic so there are 2n~l acyclicorientations Stanley (1973) showed that in general the number of acyclicorientations of F is the absolute value of C(F mdash1)

9i Recursive families and chromatic roots As was remarked at thebeginning of this chapter the location of the zeros of a chromatic poly-nomial is a fundamental problem because it subsumes the problem offinding the chromatic number One of the few positive results in thisdirection is that the zeros for some families of graphs lie near certaincurves in the complex plane Biggs Damerell and Sands (1972) defineda recursive family of graphs Tn to be a sequence of graphs in which thepolynomials C(Tn u) are related by a linear homogeneous recurrencein which the coefficients are polynomials in u In this case C(Tn u) canbe expressed in the form

fe

where the functions a and Aj are not necessarily polynomials For


example the ladders form a recursive family and as in 9c we have

a(u) = 1 02(11) = u - 1 as(u) =umdashl a4(u) = u2 - 3u + 3

Ai(u) = u2 mdash Zu + 3 A2(laquo) = 3 - u Xaiu) = 1 - u A4(u) = 1

Define a chromatic root of the family Fn to be a complex number (for which there is an infinite sequence (un) such that un is a zero ofC(Tnu) and limun = pound Beraha Kahane and Weiss (1980) obtainednecessary and sufficient conditions for ( to be a chromatic root andRead (1990) explained how their results confirm empirical observationsof Biggs Damerell and Sands concerning the chromatic roots of theladder graphs It turns out that the chromatic roots of the ladders are0 1 together with the points lying on parts of two quartic curves andthe line Hu-2

9j Planar graphs It is clear that the integers 0123 are zeros ofC(T u) for suitable planar graphs F but the four-colour theorem tellsus that the integer 4 is never a zero The first result about non-integralzeros was obtained by Tutte (1970) He observed that there is often azero close to (3 + v5)2 = 26180 and he proved that for any graphF with n vertices which triangulates the plane

mdash n~5

C(F (3 bull

Figure 4 the iterated octahedron

However this does not imply that (3 + 5)2 is a chromatic root ofevery family of plane triangulations For example the iterated octahe-dron An (Figure 4) has chromatic polynomial

C(An u) = u(u - l)(u - 2)(u3 - 9laquo2 + 29M - 32)n

Tuttes result applies to this family and consequently all the graphshave a zero near 26180 But this zero is a constant 25466independent of n there is no zero which tends to 26180 as n mdash 00


9k Chromatic roots of planar graphs Tuttes result (9j) led to specula-tion concerning the numbers bn = 2 + 2cos(27rn) as chromatic roots offamilies of planar graphs based on the observations that b = 0 63 = 164 == 2 65 = 26180 b6 = 3 and bn -gt 4 Beraha and Kahane (1979)proved that 4 is indeed a chromatic root of a family of planar graphs andBeraha Kahane and Weiss (1980) proved the the same thing for 6567and 610 Concerning the numbers bn in general Tutte (1984) observesthat their significance is not yet properly understood

91 Zero-free intervals If F is a graph which triangulates the planethere are no non-integral zeros pound of C(T u) for which C lt 25466 thezero of the octahedron mentioned in 9j (Woodall 1992)

9m Confluence of the deletion-contraction method In the method ofdeletion and contraction we are free to choose any edge at each stepThe fact that the order of choosing edges does not affect the final resultis obvious given the concrete interpretation of the chromatic polynomialin terms of colourings However the deletion-contraction method maybe applied formally as a set of so-called rewriting rules and then it isnecessary to prove that there is a normal form independent of the orderin which the rules are applied This follows from two general propertiesof the rewriting rules known as well-foundedness and local confluence(Yetter 1990)

9n The umbral chromatic polynomial If P is a partition of an n-set inwhich there are a parts of size i then we define the formal expression

Given a graph F let

pwhere the sum is over all colour-partitions P of VT with r parts Clearlyputting ltfgti mdash ltfgt2 mdash bull bull bull = (jgtn-i = 1 we obtain the ordinary m r(F) asdefined on p 63

Ray and Wright (1992) show that the corresponding generalization ofthe chromatic polynomial is obtained by replacing the expressions U(r)by what are known as the conjugate Bell polynomials bfu) Thus theydefine the umbral chromatic polynomial

r=lThey obtain interpolation formulae like those in 9g and analogues ofother properties of the ordinary chromatic polynomial

10

Subgraph expansions

It is clear that calculating the chromatic polynomial of a graph is atleast as hard as finding its chromatic number The latter problem isknown to be difficult in a technical sense which appears to correspondwith practical experience (More details may be found in the AdditionalResults at the end of Chapter 13) There are nevertheless good reasonsboth theoretical and practical for studying methods of calculating thechromatic polynomial which are more sophisticated than those discussedin the previous chapter These methods are based on the idea of anexpansion in terms of certain subgraphs

Definition 101 The rank polynomial of a general graph F is thefunction defined by

R(Txy) = Er

scErwhere r(S) and s(S) are the rank and co-rank of the subgraph (5) offIf we write R(T x y) mdash pound prsx

rys then prs is the number of subgraphsof F with rank r and co-rank s and we say that the matrix (prs) is therank matrix of T

For example the rank matrix of the graph K3gt3 is193684 9117 45 6

L 81 78 36 9 1J


Here the rows are labelled by the values of the rank r from 0 to 5 and thecolumns are labelled by the values of the co-rank s from 0 to 4 We noticethat since r(S)+sS) mdash S for all S C ET an antidiagonal (sloping frombottom left to top right) corresponds to subgraphs with a fixed numbert of edges and consequently sums to the binomial coefficient (^) Weobserve also that the number in the bottom left-hand corner (generallyPn-io where n is the number of vertices) is just the tree-number of thegraph These facts mean that in this case very few entries need to becalculated explicitly

As we shall see several interesting functions can be obtained by as-signing particular values to the indeterminates x and y in the rank poly-nomial Trivially putting y = x gives R(Txx) = (x + l)^1 becauserS) + s(S) = S for all S C ET The main result to be proved inthis chapter is that by assigning certain values to x and y we obtain thechromatic polynomial

For any natural number u let [u] denote the set 12 u whichwe shall think of as a set of u colours and let [u]x denote the set of allfunctions ugt X mdashraquo [u] For a general graph F the set [u]vr containssome functions which are vertex-colourings of F with u colours availableand some functions which are not vertex-colourings since they violate thecondition that adjacent vertices must receive different colours In orderto pick out the vertex-colourings we make the following definition

Definition 102 For each w euro [v]vr we define the indicator function

QET- 01 as follows

_ f 1

In particular cD(e) = 0 if e is a loop

Lemma 103 IfT is a general graph and u is a natural number then

~ _ f 1 if e has vertices vV2 such that w(vi)0 otherwise

nProof The product n ^ ( e ) s z e r 0 unless Q(e) = 1 for all e euro EFand this is so only if ugt is a vertex-colouring of F Thus the sum of theseproducts is the number of vertex-colourings of F using at most u coloursThe result follows from Proposition 92 bull

Theorem 104 The chromatic polynomial of a graph F with n verticeshas an expansion in terms of subgraphs as follows

C(Tu)=SCET

Subgraph expansions 75

Proof For any natural number u we have

^ e ) = EExpanding the product of terms 1 + (e) we obtain a sum ofexpressions n(e)gt o n e fdegr e a c n subset S C ET That is

c(rlaquo)= X) E

We now switch the order in the double sum For each 5 C ET letVS mdash V(S) then any function from VS to [u] is the restriction to VSof u l v r v s l functions from VT to [u] Thus

E E n(-(laquo)-)= E-|yrxvsi E n w - 1 ) w6[u]vr scsrees sccr ue[u]vs ces

Consider the product n ( ^ ( e ) ~ 1) o v e r a ^ edges e euro 5 If the productis non-zero uj(e) must be 0 for each e G S which means that w isconstant on each component of (S) In this case the value of the productis (mdash1)ISL If S) has c components there are uc such functions ui hencethe sum of the product over all v)vs functions ugt VS mdashgt [u] is (mdashl)slufiThe result follows from the equation

|VT VS + c = n - VS + c = n- rS)

D

Corollary 105 The chromatic polynomial and the rank polynomialof a general graph T with n vertices are related by the identity

If the chromatic polynomial is

C(T u) = boun + hun-1 ++ bnû + bn

then the coefficients bi can be expressed in terms of the entries in therank matrix as follows

j

Proof The identity between the polynomials follows directly fromTheorem 104 and the definition of the rank polynomial In terms of the


coefficients we have

M = C(Tu) = unR(T -u~ -1)

Equating coefficients of powers of u and rearranging the signs we havethe result stated above bull

The formula for the coefficients expresses bi as an alternating sum ofthe entries in the zth row of the rank matrix This formula was firststudied by Birkhoff (1912) in the original paper on chromatic polyno-mials and Whitney (1932a) For example from the rank matrix for^33 given above we have

bx = - 9 62 = 36 h = -84 + 9 = -75 b4 = 117 - 45 + 6 = 78

65 = -81 + 78 - 36 + 9 - 1 = -31

This checks with the result obtained in Chapter 9 by a different method

3t3 u)=u6 - 9M5 + 36M4 - 75u3 + 78u2

Proposition 106 Let F be a strict graph of girth g having m edgesand r cycles of length g Then with the above notation for the coeffi-cients of the chromatic polynomial of T we have

(1) (-l)lt6-=(7) for i = 0lg-2

(2) ( - l ) raquo - V i = ( f l - i ) - -

Proof A subgraph of F with rank i lt g mdash 2 must have co-rank zerosince F has no cycles with fewer than g edges Thus for alH lt g mdash 2we have p^ = (trade) and ptj = 0 if j gt 0 Further the only subgraphsof F with rank g mdash 1 are the ( tradeJ forests with g mdash 1 edges (which haveco-rank zero) and the r cycles with g edges (which have co-rank 1)Thus

Pg-io = ( ) pg-ii=ri pg-u=--0 if j gt 1

The result follows from the expression for the coefficients of the chro-matic polynomial bull

We observe that for a strict graph the girth g is at least 3 so the coef-ficient of un~1 in the chromatic polynomial is mdash m where n and m arethe numbers of vertices and edges respectively


The formula for the coefficients of the chromatic polynomial is analternating sum and its use involves counting many subgraphs whichcancel out in the final result Whitney (1932a) discovered a reductionwhich involves counting fewer subgraphs His result also shows that thenon-zero coefficients of the chromatic polynomial alternate in sign thatis (mdashl)lbi is always positive Let F be a simple graph whose edge-setET = e e2 em is ordered by the natural order of subscripts Thisordering is to remain fixed throughout our discussion A broken cyclein F is the result of removing the first edge from some cycle in otherwords it is a subset B of ET such that for some edge e we have

(1) B U e is a cycle in F (2) i gt I for each edge e G B

The next proposition expresses the coefficients of the chromatic polyno-mial in terms of the subgraphs which contain no broken cycles clearlysuch subgraphs contain no cycles and so they are forests

Proposition 107 (Whitney 1932a) Let T be a strict graph whoseedge-set is ordered as above and let C(Tu) = ~^2biUn~l Then (mdash l)lbiis the number of subgraphs ofT which have i edges and contain no brokencycles

Proof Suppose Bi 52gt bull bull bull Bt is a list of the broken cycles of F indictionary order based on the ordering of ET Let raquo (1 lti ltt) denotethe edge which when added to Bi completes a cycle The edges arenot necessarily all different but because of the way in which the brokencycles are ordered it follows that j is not in Bj when j gt i

Define So to be the set of subgraphs of F containing no broken cycleand for 1 lt h lt t define S^ to be the set of subgraphs containing Bhbut not Bh+iBh+2 bull bull bull Bt- Then So S i S^ is a partition of theset of all subgraphs of F We claim that in the expression

the total contribution to the sum from S i St is zeroSuppose 5 is a subset of ET not containing ^ then S contains Bh if

and only if S U fh contains Bh Further S contains Bi (i gt h) if andonly if 5 U fh contains Bi since fh is not in Bi Thus if one of thesubgraphs S) S U fh) is in S then both are in S^ They have thesame rank but their co-ranks differ by one and so their contributionsto the alternating sum cancel Consequently we need only consider thecontribution of So to J2(-iyPijbull Since a subgraph (S) in So is a forestit has co-rank j = 0 and rank i = S whence the result bull

Corollary 108 Let T be a strict graph with rank r Then the co-


efficients of C(Tu) alternate strictly in sign that is (mdashl)lbi gt 0 fori = O l r

Proof The characterization of Proposition 107 shows that (mdashl)6j gt0 for 0 lt i lt n In order to obtain the strict inequality we must showthat there is a subgraph with i edges and containing no broken cycle for1 = 0 1 r Suppose we successively remove edges from F in such away that at least one cycle is destroyed at each stage this process stopswhen we reach a subgraph (F) of F with F = r and s(F) = 0 Let usorder the edges of F so that the edges in F come first Then (F) containsno broken cycle and any subset of F generates a subgraph containingno broken cycle Thus we have produced the required subgraphs andthe result follows bull

Recall that at the beginning of Chapter 9 we observed that 6j = 0 ifi mdash nnmdash1 nmdash(cmdash1) where n = VT and F has c components Thatis bi = 0 if i = r + 1 n Thus we have shown that the coefficientsof the chromatic polynomial alternate strictly and then become zero

Additional Results

10a Inequalities for the coefficients of the chromatic polynomial If Fis a connected strict graph with n vertices and m edges and C(F u) =S6jUnJ then

10b Codichromatic graphs An example of two non-isomorphic generalgraphs having the same rank matrix was found in the 1930s by MarionC Gray (see Figure 5)

Figure 5 two general graphs with the same rank matrix


Tutte (1974) drew attention to this work and constructed pairs of strictgraphs which have the same rank matrix

10c V-functions A function defined on isomorphism classes of graphsand taking values in a ring A is a V-function if it satisfies the followingconditions(a) If F is empty then f(T) = l(b) If T is the union of disjoint graphs Fj F2 then (F) = ( r i ) ( r 2 ) (c) If e is any edge of F which is not a loop then satisfies the deletion-contraction formula

It is easy to see that the chromatic polynomial and the rank polyno-mial suitably normalized are V-functions The most general V-functionis constructed as follows For any sequence i = i o i i i 2 of non-negative integers with finite sum let v(T i) be the number of spanningsubgraphs of F which have iamp components of co-rank k for k gt 0 Lets = (SQ SI laquo2 bull bull bull) be any infinite sequence of elements of A and let

Then s is a V-function and every V-function can be written in thisway (Tutte 1947b)

lOd The rank polynomial as a V-function By taking the ring A to bethe ring of polynomials with integer coefficients in two indeterminatesxy and s to be the sequence defined by s = xyl we obtain the rankpolynomial (with suitable normalization) as a V-function

lOe Homeomorphic graphs The operation of replacing an edge withends u v by two edges with ends u w and w v where w is a new vertexis known as subdividing the edge Regarding a graph as a topologicalspace in the obvious way it is clear that subdividing an edge results ina graph homeomorphic to the original one In general two graphs arehomeomorphic if they can both be obtained from the same graph bysequences of subdivisions A graph function is said to be a topologicalinvariant if its values on homeomorphic graphs are equal It can beshown that a non-trivial V-function is a topological invariant if andonly if ( ) = - 1

lOf Interaction models The formula obtained in Lemma 103 arisesnaturally in theoretical physics The vertices of the graph F = (V E)


are particles each of which which can have one of u attributes so thata state of the system is a function UJ V mdashgt [u] Each pair of adjacentvertices corresponding to an edge e amp E has an interaction ^(e) whichdepends on the state w and the weight I(w) is the product of theinteractions iuj(e) The partition function is the sum of all the weights

The chromatic polynomial is the special case arising when iu is theindicator function a as in Definition 102 that is iue) is 1 if the endsof e have different attributes and is 0 otherwise In general the valueof u and the function iu determine an interaction model An account ofthe properties of such models was given by Biggs (1977b)

lOg The Ising and Potts models Using the interaction model termi-nology suppose iu (e) is a if the ends of e have the same attribute in thestate w and 1 otherwise For general u this is known as the Potts modeland the special case u = 2 is known as the Ising model The partitionfunction for the Potts model can be expanded as a rank function

pound a - 1)SCEV U

lOh A general form of the subgraph expansion An interaction modelfor which iu(e) takes only two values one when the ends of e havethe same attribute and another when they have different attributesis said to be a resonant model The expansions in terms of the rankfunction described above can be generalized to any resonant model inthe following way Let F and G be resonant models for which the two(distinct) values of iu(e) are i o and gigo respectively and let fa =19i +6 i = 01 Then Zp(T) can be expanded in terms of the values ofZlt3 on the subgraphs of F as follows

lOi Another expansion of the chromatic polynomial Nagle (1971) ob-tained the following expansion

C(Tu)= ] T u l v r v s l ( l -laquo- 1 ) l B r s l iVlaquo5gtlaquo) seer

where the function N has the following properties (a) if F has an isth-mus then iV(F u) = 0 (b) TV is a topological invariant

11

The multiplicative expansion

In this chapter and the next one we shall investigate expansions of thechromatic polynomial which involve relatively few subgraphs in compar-ison with the expansion of Chapter 10 The idea first appeared in thework of Whitney (1932b) and it was developed independently by Tutte(1967) and researchers in theoretical physics who described the methodas a linked-cluster expansion (Baker 1971) The simple version givenhere is based on a paper by the present author (Biggs 1973a) There areother approaches which use more algebraic machinery see Biggs (1978)and lie

We begin with some definitions Recall that if a connected graph F isseparable then it has a certain number of cut-vertices and the removalof any cut-vertex disconnects the graph A non-separable subgraph of Twhich is non-empty and maximal (considered as a subset of the edges)is known as a block Every edge is in just one block and we may thinkof F as a set of blocks stuck together at the cut-vertices In the caseof a disconnected graph we define the blocks to be the blocks of thecomponents It is worth remarking that this means that isolated verticesare disregarded since every block must have at least one edge

Let Y be a real-valued function defined for all graphs and having thefollowing two properties

PI Y(T) = 1 if T has no edgesP2 Y(T) is the product of the numbers Y(B) taken over all blocksB ofF


Given such a Y let X be the real-valued function defined by

X(T) - ]T YS)SCET

An example of a function satisfying PI and P2 is obtained by takingY(T) = xr(r^ys(r where x and y are a given pair of real numbers inwhich case the corresponding X is (an evaluation of) the rank polyno-mial of F The fact that this Y satisfies P2 is a consequence of theequations

r(r) = poundgt(pound) (r) = poundgt(pound)where the sums are taken over the set of blocks B of F

Lemma 111 If the function Y satisfies P I and P2 then the corre-sponding function X satisfies the same properties

Proof (PI) If r has no edges then the sum occurring in the definitionof X contains only one term F(0) which is 1

(P2) Suppose F has just two blocks Fi and F2 with edge-sets E andE2 Then for any S C ET the sets Si = S fl Ex and S2 = S D E2 aresuch that S = St U S2 and St n S2 = 0 Thus the blocks of (S) in Fare the blocks of (Si) regarded as a subgraph of Fi together with theblocks of (^2) regarded as a subset of F2- By P2 we have

Y(S)r = Y(S1)r1Y(S2)r2-

(This equation remains true if either or both of Si S2 are empty byPI) Consequently

X(T) = ]T Y(S)r = Y ESCET SlCE1S2CE2

S1CE1 S2CE2

If F has b gt 2 blocks we have a similar argument taking Ti to be thefirst 6mdash1 blocks Hence the general result follows by induction bull

We shall now transform the sum X(T) into a product using exponen-tial and logarithmic functions We require also the fundamental identityunderlying the principle of inclusion and exclusion that is

ICJ

provided that J is not the empty set

Definition 112 Let (X Y) be a pair of functions as above and sup-pose that the values of X are positive Then the logarithmic transform

The multiplicative expansion 83

of the pair (X Y) is the pair of functions (X Y) denned by

X(T) = (-l)lpound r l 5 2 (-1)11 logX5) Y(T) = expX(F)SCET

Proposition 113 Let F be a general graph IfT has no edges or ifF is separable and has no isolated vertices then X(T) = 0

Proof If ET is empty then X(T) = 1 and consequently X(T) = 0Suppose that F has no isolated vertices and is separable Then eitherit is disconnected or it is connected and has at least one cut-vertex Ineither case it can be expressed as the union of two subgraphs (E) and(E2) with E and E2 non-empty and disjoint For S C poundT we have

X(S) = X(S1)X(S2)

where Si = S n E and S2 = S n E2 and so logX(S) = loglog X(S2- This justifies the following calculation

SCET

5Z E (-l)|Sll+|S2|(log A-lt5) + logSÊj S2Cpound2

(-l)lsllogX51) 52 (-x S 2 C pound 2

52 (-i)|S2|iog x(s2

Both E and pound 2 are non-empty so the fundamental inclusion-exclusionidentity stated above implies that the entire expression is zero and wehave the result bull

Theorem 114 Let F be a non-separable graph and let (X Y) be thelogarithmic transform of the pair (X Y) Then X(T) has a multiplicativeexpansion

XT)= n YS)SCEF

in which Y is equal to 1 (and so may be ignored) for separable subgraphsofT

Proof The fact that YS) = expX(S) = 1 for separable subgraphs(S) follows from the previous lemma since by definition a subgraphhas no isolated vertices


We shall prove that

iogx(r)=SCET

from which the theorem follows by taking exponentials Now from thedefinition of X

E x(S)=SCEV scEr RCS

and (R) as a subgraph of (5) is identical with (R) as a subgraph of TWriting Y = SR the right-hand side becomes

E E (-D|iJ|+|y|(-i)|fl|iogx(igtRCET YCEVR

= E ^gx(R) E (-D|v|-RCEV YCEVR

The inner sum is non-zero only when EFR = 0 that is when R = ETThus the expression reduces to log X(ET) = ogX(T) as required

bull

We now apply the general theory of the logarithmic transform to theparticular case of the chromatic polynomial We take the function Y tobe

This satisfies PI and P2 and by Theorem 104 the corresponding Xfunction is

Lemma 115 Let (XUYU) denote the particular pair of functionsgiven above Then for a given graph T Xu(Y)and YU(T) can be definedfor all sufficiently large integers u

Proof In order to define XUT) satisfactorily we must ensure thatogXu(S) is denned for all subsets S C poundT Now if u is an integergreater than the chromatic number of F it is clear that C((S)u) ispositive and so the logarithm of XU(S) = u~^v^s^C((S)u) is defined

bull

We can eliminate the logarithmic and exponential functions from thegeneral definition of Y obtaining

Y(T)= I ] ltSraquoe(S) where e(5) = (-l)lpound r s | SCET


For the particular case Yu we get

YU(T)= JJ u-^C((S)u)y(sscEr

which is valid for all sufficiently large positive integers u The productformula shows that Yu is a rational function in its domain of definition

We shall find it convenient to deal separately with the Yu functionfor a single edge that is YU(K2)- It is easy work this out explicitlyYu(K2) = -vTl

Proposition 116 For every non-separable graph A having more thanone edge there is a rational function q(A u) such that the chromaticpolynomial of a graph T has a multiplicative expansion

where the product is taken over all those non-separable subgraphs AoFwhich have more than one edgeProof We have seen that if (XUYU) is the pair defined by

Yu(r) = (-1)1-^) XU(T) = u^vrC(T u)then Yu is defined Jbr all subgraphs of F provided u is a^sufficientlylarge integer and YUK2) = (1 mdash M1) Setting q(Au) = YUA) whenEA gt 1 and applying Theorem 114 we see that the identity holds foran infinite set of values of u Since both sides are holomorphic functionsthey are identical bull

The functions q(T u) can be found explicitly for certain standardgraphs For example for the cycle graph Cn the only subgraph occuringin the product is Cn itself hence

C(Cnu)=un(l-u-x)nq(Cnu)

By a result of Chapter 9 the left-hand side is (u - l )n + (-l)n(u - 1)so that

This simple calculation highlights an apparent circularity which arisesif we propose to use the multiplicative expansion to calculate chromaticpolynomials The difficulty is that the right-hand side of the multiplica-tive expansion of C(T u) contains a term q(Tu) and we have as yetno way of finding q(T u) without prior knowledge of C(Tu) In thenext chapter it will be shown that this seemingly fundamental objectioncan be surmounted by means of a few simple observations We shall alsoobtain a version of Proposition 116 in which the number of subgraphsinvolved is reduced still further


Additional Results

l l a The q function of a crossed cycle Let C+ denote a graph con-structed from the cycle graph Cn by the addition of one edge joiningtwo distinct vertices which are not adjacent in Cn Then

l i b Theta graphs (Baker 1971) Let Qrst denote the graph consistingof two vertices joined by three disjoint paths of length r s and t copyrsthas n = r + s + tmdash 1 vertices and r + s + t edges and q(Qrstu) is

1 - (1 - u)r-n - (1 - u)s~n - (1 - it)- + (2 - u)(l - u)~n

(1 - (1 - u)r-n)(l - (1 - u)s~n)(l - (1 - uf-n)

l i e The multiplicative expansion of the rank polynomial If Y(T) =xr(r)y(r) then X(T) = R(T x y) and the logarithmic transform appliedto the pair (X Y) leads to a multiplicative expansion

where the product is over all non-separable subgraphs A of F which havemore than one edge (Tutte 1967)

l i d Whitneys theorem on counting subgraphs In Chapter 10 we ob-tained a formula for the coefficients of the chromatic polynomial whichinvolved counting all the subgraphs In this chapter we have shownthat in theory only the non-separable subgraphs are needed Whitney(1932b) obtained this result in a different way by showing that there is ageneral expression for the number of subgraphs of any particular type interms of the numbers of non-separable subgraphs Specifically let fit(F)be the number of subgraphs of F which have a given type t where atype is determined by the number of blocks of each isomorphism classThen there is a polynomial function ltJt independent of F with rationalcoefficients and no constant term such that

nt(r)=t(nlty(r)nT(r) )where a T are the nonseparable types with not more edges than tFor example if A|| denotes the type with one block isomorphic to Kzand two blocks isomorphic to K2 we have

7 1 2

2 +


where 0 is the type of the theta graph copy221 and the other notation isself-explanatory

l i e An algebraic framework In order to unify the theory of the mul-tiplicative expansion and Whitneys theorem described above Biggs(1977b 1978) introduced the following algebraic framework Define Stthe set of star types to be the set of isomorphism classes of non-separablegraphs and Gr the set of graph types to be the set of functions from Stto the non-negative integers with finite support Let X and Y respec-tively be the vector spaces of real-valued functions defined on St andGr When St is regarded as a subset of Gr in the obvious way we havea projection J Y mdashgt X

For a given graph F of type g define cg euro Y by the rule that cg(t) isthe number of subgraphs of F which are of type t Then Jcg representsthe numbers of non-separable subgraphs of F Whitneys theorem assertsthat there is an operator W X mdashbull Y such that

W(Jcg) = cg for all g euro Gr

In the papers quoted it is proved that W = B~1U where B is alinear operator defined by a certain infinite matrix and U X mdash Y isthe monomial mapping defined by

l l f Expansions as linear functional Denote the subspaces of X andY consisting of vectors with finite support by Xo and Yo respectivelyThe real vector spaces Xo and Yo admit scalar products defined in theusual way

((xix2raquo = 53XI(ltT)X2(ltT) (yiyz) = 53yi()y2()-a t

For any given m euro Yo there is a linear functional M defined by M(y) =(y m) On vectors cg representing real graphs Mcs) is by definitionof the scalar product a sum over subgraphs in which each subgraph oftype t contributes m(i) In the authors papers quoted above it is shownthat under certain conditions there is a corresponding linear functionalC on Xo such that

expC(Jcg) = M(cg) for all g e Gr

Explicitly we have

pound(x) = ((x 1)) where 1 =


l l g The Hopf algebra framework There is clearly a substantial amountof algebraic structure underlying Whitneys theorem and the multiplica-tive expansion Schmitt (1993) carries this idea to its logical conclusionby introducing coalgebras and Hopf algebras He shows that the algebraof formal power series with rational coefficients over St can be given thestructure of a Hopf algebra and that it is isomorphic to the dual of thefree module with rational coefficients over Gr Whitneys theorem is adirect consequence of the isomorphism

Another approach using Hopf algebras is discussed by Ray (1992)

12

The induced subgraph expansion

In this chapter we shall modify the multiplicative expansion of the chro-matic polynomial in such a way that the induced subgraphs are theonly ones occurring in the formula This procedure has two advantagesFirst there are fewer induced subgraphs than subgraphs in general andsecondly the function which takes the place of the q function (in thenotation of Proposition 116) turns out to be trivial for a wider class ofgraphs

The formal details of the transition to induced subgraphs are quitestraightforward For any non-separable graph A define

Q(Alaquo)=JIg(Ati)where the product is over the set of spanning subgraphs A of A thatis those for which VA mdash VA It follows immediately that Q is a ratio-nal function of u For example the cycle graph Cn has just one non-separable spanning subgraph which is Cn itself Thus the definition ofQ gives

Q(Cnu) = q(Cnu) =

Proposition 121 The chromatic polynomial has a multiplicative ex-pansion

where the product is over all non-separable induced subgraphs ofT havingmore than one edge


Proof The factors which appear in Proposition 116 can be groupedin such a way that each group contains those subgraphs of F which havea given vertex-set This grouping of factors corresponds precisely to thatgiven in the definition of Q and the resulting expression for C since eachsubgraph A of F is a subgraph of exactly one induced subgraph A of F(the one for which VA = VA) and conversely each subgraph of A is asubgraph of F bull

The crucial fact which makes the multiplicative expansion useful inpractice is that the q and Q functions are rational functions of a specialkind Specifically it can be shown that

where v and 6 are polynomials whose degrees satisfy

deg S - deg v gt VT - 1

The first satisfactory proof of this important fact was given by Tutte(1967) using the notion of tree mappings An algebraic proof wasgiven by Biggs (1978 see also l i e and l lf)

Given this result we can prove the same thing for Q

Proposition 122 LetT be a non-separable graph ThenQ(Tu) maybe written in the form

where v and 6 are polynomials such that deg 6 mdash deg v gt VT mdash 1

Proof The function Q is defined to be the product of functions q overa set of graphs with the same number of vertices Thus the result for qimplies the result for Q bull

We are now in a position to overcome the circularity mentioned atthe end of the previous chapter It is possible using Proposition 122to calculate both C(F u) and Q(T u) provided only that we know theQ functions for all proper induced subgraphs of F that is the inducedsubgraphs not including F itself To see this we write the formula ofProposition 121 as

where P(u) is a product of rational functions corresponding to the properinduced subgraphs including the vertices (for each of which we havefactor u) and the edges (for each of which we have a factor 1 mdash u~1)

The induced subgraph expansion 91

It follows that P(u) can be written as a polynomial of degree n mdash VTplus a power series in u~l

P(u) =un + a i u n - 1 + + a n _ iu + an + ctn+ivT1 +

But following Proposition 122 the function Q(F u) can be written

Q(F u) = 1 + (30u~n+l + l U - n +

It follows that multiplying P(u) by this expression does not alter thecoefficients of unun~1 u2 in P(u) Thus the polynomial part ofP(u) is a correct expression for C(T u) except for the coefficients of uand 1 But these coefficients in C(F u) are easily found by noting thatu(u - 1) is a factor of C(T u) It follows that both C(T u) and QT u)are determined by the known function P(u)

An example will elucidate this argument Take F = K then the onlyproper induced subgraphs of T having more than one edge are the fourcopies of K3 = C3 Thus

Q(K4u)

Dividing (u - I)2 into u2(u - 2)4 gives P(u) = u4 - 6u3 + llu2

and so

C(K4 u) = u4- 6u3 + llu2 - au + b

Since u(u mdash 1) is a factor of C(K4 u) it follows that a = 6 6 = 0 and

C(K4 u) = u4 - 6w3 + llu2 - 6u = u(u - l)(u - 2)u - 3)

We can also find QK4 u) by substituting back obtaining2 u - 3

The technique which we have just described has the important conse-quence that we can calculate chromatic polynomials merely by countinginduced subgraphs without knowing any C and Q functions in advanceIn particular it implies that the chromatic polynomial is reconstructiblein the sense of 7f

To make this explicit suppose that Ai A2 A is a list of the iso-morphism types of non-separable induced subgraphs of F where K =Ai and K2 = A2 axe included for the sake of uniformity and F = AThen we define a matrix N = (n^) by putting mj equal to the numberof induced subgraphs of Araquo which are isomorphic with Aj We may sup-pose that the list has been ordered in such a way that N is a triangularmatrix each of whose diagonal entries is +1


Proposition 123 The matrix N completely determines the chromaticpolynomial ofTProof We know the C and Q functions for all the graphs with atmost three vertices Now suppose we know the C and Q functions forthe induced subgraphs of F with at most t vertices then we can findthe C and Q functions for each induced subgraph with t + 1 verticesby using the technique previously explained Thus using this procedurerecursively leads to the chromatic polynomial of F bull

For example the following is a complete list of the non-separableisomorphism types of induced subgraphs of the ladder graph L3 (Thegraph itself occurs as A6 in Figure 6)

A Aa A A5

Figure 6 the induced subgraphs of L3

The N matrix for F isri2345

6 l j

To see how the method works suppose that we have completed thecalculations for subgraphs with at most four vertices The C and Qfunctions for these graphs are as follows

A2 A3 A4

C u ( u - l ) u(u - l)(u2 - 3w + 3)

Q (u - l)u u(u - 2)(u - I)2 u(u2 -3u + 3)(u - I)3

The remainder of the calculation now proceeds in the following way Wehave C(A5u) = P5(u)Q(A5u) where

= u(u - l)(u - 2)(w2 - 3w + 3)


Here (atypically) -Ps(w) is a polynomial divisible by u(u mdash 1) and so

C(A5u) = u(u- l)(u - 2)(u2 -3u + 3) and Q(A5u) = l

At the next stage we have C(Asu) = PQ(u)Q(Aeu) where

(I)6

= u6 - 9u5 + 34u4 - 67u3 + 67w2

Here Pe(w) is not a polynomial Extending the terms in u2 and aboveto a polynonial divisible by u(u mdash 1) we get C(T u) = u6 mdash 9u5 + 34u4 mdash67u3 + 67u2 - 26u

One noteworthy feature of the preceding calculation is that Q(A$ u) =1 although A5 is a non-separable graph This means that we couldhave ignored A5 completely both in setting up the matrix N and inthe subsequent calculations The next proposition shows that there is alarge class of non-separable graphs T for which Q(F u) = 1

Proposition 124 (Baker 1971) the graph T is quasi-separable inthe sense of Definition 98 then Q(Tu) = 1

Proof We prove this result by induction on the number of verticesof F The result is true for all quasi-separable graphs with at mostfour vertices For this set contains only one graph (the graph shown inFig3 p 67) which is not in fact separable and the claim can be readilychecked for that graph

Suppose that the result is true for all quasi-separable graphs with atmost L vertices and let T be a quasi-separable graph with L+1 verticesWe have a quasi-separation (Vi^) of T where (Vi (~l V2) is completeand (VT - (V n V2)) is disconnected The expansion of Proposition 121can be written in the form

) = P(u)Q(Tu)

where P(u) is a product of factors corresponding to the proper non-separable induced subgraphs of F If U is any proper subset of VTfor which U V and U V2 then (U) is a quasi-separable graphwith quasi-separation (Vi n U) (V2 CiU) By the induction hypothesis

Thus the non-trivial terms in the product P(u) correspond to thesubsets of V and the subsets of V2 However a subset of V D V2 occursjust once rather than twice It follows that

P(u) =C((V1DV2u)


Since Proposition 99 tells us that C(T u) is also equal to this expressionit follows that Q(F u) = 1 and the induction step is verified bull

We observe that the graph A5 in the example preceding the propo-sition is in fact quasi-separable and so the fact that Q(A5u) = 1 isexplained

The following theorem is the culmination of the theory developed inChapters 10-12

Theorem 125 The chromatic polynomial of a graph is determinedby its proper induced subgraphs which are not quasi-separable

Proof This theorem follows from Propositions 121 and 124 bull

We close this chapter with a brief explanation of how the theory can beused to study the chromatic polynomial of an infinite graph SupposeP is an infinite graph which can be regarded in some way as the limit ofa sequence of finite graphs 9n with (FvEnl = vn say The appropriatedefinition of the chromatic polynomial of ltfr is

provided the limit exists for a suitable range of values of u In theoreti-cal physics this is known as taking the thermodynamic limit and someexistence results have been proved for interaction models as defined inlOf Grimmett (1978) obtained strong results for the rank polynomialbut for our present purposes blind faith and ignorance will suffice

If bdquo has reasonable regularity properties then the number of inducedsubgraphs of a given type in tn is avn where a is a constant represent-ing the density that is the number of induced subgraphs of that typeper vertex For example if ampn is regular of degree k the number ofedges is (k2)vn and so the density of edges is k2 If we now take the(lwn)th root of the multiplicative Q-formula for C(lra u) we get a termu (corresponding to the vertices) a term (1 mdash u~x)k2 (corresponding tothe edges) and in general a term QA u)a for each induced subgraphA of density a This leads to a definition of the chromatic polynomialwhich does not depend on the approximating sequence ampn Unfortu-nately nothing is known about the convergence of the infinite productalthough it is clear that the smallest induced subgraphs which are theeasiest to count contribute the largest terms

A good illustration is provided by the infinite plane square latticegraph Here the only induced subgraphs which are not quasi-separableand have not more than eight vertices are the vertices edges CVs


and Css with densities 121 and 1 respectively It follows that anapproximation to C^ in this case is

The correct value when u = 3 is known to be (43)32 = 1540 (Lieb1967) whereas the approximation gives 1512 For larger values of uit seems likely that the approximation is better but no general resultsare known (See also 12f)

Additional Results

12a The Q function for complete graphs We have

Q(Knu)= 1Q (u-i)(i0ltiltn-l

where ) = (-ir-1-(T1)

12b The Q functions for all graphs with less than six vertices The onlygraphs with less than five vertices which are not quasi-separable are KltiK3 K4 and d and we have already found Q for all these WritingQ(T u) = 1 4- r(T u) the r functions are as follows

r(K2u) = 1u

r(K3u)= -lu-lf

r(K4 u) = -(2u - 3)u(u - 2)2

r(C4u)= l (w- l ) 3 -

The relevant graphs with five vertices are K$ W5 W~ (the wheel withone spoke removed) X2gt3 and C5 The r functions are

r(K5 u) = -(6u4 - 48w3 + 140u2 - 176u + 81)(u - l)4(u - 3)4

r(W5 u) = (3u2 - 9u + 7)u(u - 2f(u2 - 3u + 3)

r(W-u) = (2u2 - 6u + 5)u(u - 2)(M2 - 3u + 3)2

r(K23 u) = (u3 - 6u2 + llu - 7)u(u2 - Zu + 3)3

r(C5u)= -l(u-l)

12c Petersens graph The only non-quasi-separable induced subgraphsof Petersens graph 0 3 have 25678910 vertices and there is one


5681012

15

1024612

112410

isomorphism class in each case The N matrix is

14 19 3 130 15 10 1

Using the method described on pp 92-93 this gives the chromatic poly-nomial of O3u(u - 1)(u - 2)(u7 - 12w6 + 67u5 - 230u4 + 529u3 - 814u2 + 775u - 352)

12d The first non-trivial coefficient in q and Q If F is non-separableand has n vertices and m edges then the coefficient of u~^n~^ in theexpression for q(T u) in descending powers of u is equal to ( - l ) m Thecorresponding coefficient in QTu) is therefore XX~1)BAgt where thesummation is over all non-separable spanning subgraphs A of F (Tutte1967)

12e Chromatic powers Let ltrm(F) denote the sum of the mth powersof the zeros of C(F u) Suppose that

where the expansion is valid for |u| sufficiently large If n(F A) denotesthe number of induced subgraphs of F which are isomorphic with A wehave

where the sum is taken over isomorphism classes of non-quasi-separablegraphs (Tutte 1967)

12f Approximations for the infinite square lattice There have beenmany attempts to determine the chromatic polynomial Coo(w) of theinfinite square lattice Biggs and Meredith (1976) obtained the estimate

hu - 3 + yu2 - 2u + 5)zUsing the transfer matrix method Biggs (1977a) obtained the bounds

2 + v ^ - 4 u + 8)lt u

Kim and Enting (1979) obtained a series approximation in terms ofx = u mdash 1) apart from a simple factor it is

+ x7 + 3xs + 4x9 + 3x10 Ux 12

- 91a15 - 261a16 - 290x17

+ 24a13

254x18

8z14

13

The Tutte polynomial

There is a remarkable relationship between the rank polynomial and thespanning trees of a graph In this chapter we shall develop this theoryby giving an explicit definition of what is known as the Tutte polynomialT(F x y) of a graph F in terms of its spanning trees and then provingan identity between the Tutte polynomial and the rank polynomial

An alternative approach to the Tutte polynomial is to define it recur-sively by the deletion-contraction property

where e is neither a loop nor an isthmus This rule together with aboundary condition (see 13c) does in fact define T completely How-ever it is not immediately obvious that the method leads to a resultwhich is independent of the order in which edges are deleted and con-tracted and it provides no insight into the remarkable properties of TFor these reasons we shall follow the constructive route given below

The definition of the rank polynomial depends upon the assignmentof the ordered pair (rank co-rank) of non-negative integers to each sub-graph we shall call such an assignment a bigrading of the set of sub-graphs If F is connected the set of subgraphs whose bigrading is(r(r)0) is just the set of spanning trees of F We shall introduce anew bigrading of subgraphs which has the property that if it is givenonly for the spanning trees of F then the entire rank polynomial of Fis determined Our procedure is based initially upon an ordering of theedge-set ET although a consequence of our main result is the fact that


this arbitrary ordering is essentially irrelevant Another consequence ofthe main result is an expansion of the chromatic polynomial in terms ofspanning trees this will be the subject of Chapter 14

We now fix some hypotheses and conventions which will remain inforce throughout this chapter The graph T is a connected general graphand ET has a fixed total ordering denoted by lt If X C ET we shalluse the symbol X (rather than (X)) to denote the corresponding edge-subgraph of F and the singleton sets x C ET will be denoted by xinstead of x The rank of T will be denoted by ro thus ro = r(T) =vr -1

If X C ET and x pound ET X then the rank of X U x is either r(X)or r(X) + 1 and in the latter case we say that x is independent of XNow if r(X) bullpound ro there will certainly be some edges of T which areindependent of X and we shall denote the first of these (in the orderinglt) by X(X) We note that since

r(Y) + s(Y) = Y for all Y C ET

we have the equations

r(X U X(X)) = r(X) + 1 s(X U X(X)) = s(X)Similarly if s(X) ^ 0 then there are some edges x for which s(X x) =s(X) - 1 and we denote the first of these by n(X) We have

r(X n(X)) = r(X) s(X M(X)) = s(X) - 1

Definition 131 The A operator on subsets of ET assigns to eachset X C ET the set Xx derived from X by successively adjoining theedges (X) X(X U X(X)) until no further increase in the rank ispossible The x operator takes X to the set XM which is derived fromX by successively removing the edges fi(X) fi(X n(X)) until nofurther decrease in the co-rank is possible

We notice the following properties of the A and x operators

XCX r(Xx) = r0 s(Xx) = s(X)

Xraquo C X r(X) = r(X) a(X) = 0

We shall exploit the obvious similarity between the two operators bygiving proofs only for one of them The first lemma says that the edgeswhich must be added to a subgraph A to form Ax can be added in anyorder (In what follows the notation x lt y will mean x lt y and x ^ y)

Lemma 132 If AQB CAX then Bx = Ax

Proof If A = Ax the result is trivial Suppose

AXA = X = xix2)xt

The Tutte polynomial 99

where xi lt x2 lt lt xt and let B = A U Y where Y C X If Y = Xthen B = Ax and Bx = Axx = Ax If Y X let xa be the first edgein X Y Then if an edge x is independent of B it follows that xis independent of A U xi xa-i (which is contained in B) and soxa lt x since xa is the first edge independent of

gtlUxi x a_i

But xa itself is certainly independent of B since when we add the edgesin X to A the rank must increase by exactly one at each step Thusxa = X(B) and by successively repeating the argument with B = B UX(B) B = BU X(B) we have the result bull

Lemma 133 IfACB and r(B) ^ r0 then X(B) e ^4A

Proof Since r(B) ^ ro there is a first edge X(B) independent of Band consequently independent of A Suppose X(B) is not in Ax Theneach edge x in Ax A must satisfy x lt X(B) and so x is not independentof B also since A C B no edge in A is independent of S Thus alledges in Ax are not independent of B and r(B) = r(Ax) mdash ro This isa contradiction so our hypothesis was false and X(B) is in Ax D

We note the analogous properties of the i operator

A C B C A =gt B = A B ltZ A and s(B) ^ 0 =raquo i(B) ^

The next definition introduces a new bigrading of the subsets of ET

Definition 134 Let X be a subset of ET An edge e in ETX is saidto be externally active with respect to X if n(X U e) = e An edge inX is said to be internally active with respect to X if X(X f) = f Thenumber of edges which are externally (internally) active with respect toX is called the external (internal) activity of X

We shall denote the sets of edges which are externally and internallyactive with respect to X by Xe and X1 respectively and use the notation

X+=XUXeuro X~=XXL

These concepts are motivated by their interpretation in the case of aspanning tree because in that case they are related to the systems ofbasic cycles and cuts which were discussed in Chapter 5

Proposition 135 For any spanning tree TofTwe have(1) the edge e is externally active with respect to T if and only if e is thefirst edge (in the ordering lt) of cyc(T e)(2) the edge f is internally active with respect to T if and only iff is thefirst edge (in the ordering lt) of cut(T )


Proof By definition e is externally active if and only if e is the firstedge whose removal decreases the co-rank of TU e But TUe containsjust one cycle which is cyc(T e) and any edge whose removal decreasesthe co-rank must belong to this cycle

The second part is proved by a parallel argument bull

Definition 136 The Tutte polynomial of a connected graph F withrespect to an ordering lt of ET is denned as follows Suppose ty is thenumber of spanning trees of F whose internal activity is i and whoseexternal activity is j Then the Tutte polynomial is

Remarkably it will turn out that T is independent of the chosen ordering

In order to obtain the main result we shall investigate the relation-ship between the concepts just defined and the following diagram ofoperators

A - ^ B

VC - ^ V

Here A denotes all subsets of ET B denotes subsets Z with r(Z) = r0C denotes subsets W with s(W) = 0 and V denotes subsets T withr(T) = ro and s(T) = 0 (that is spanning trees) It is worth remarkingthat the diagram is commutative although we shall not need this result(see 13g)

Proposition 137 Let X be any subset in the image of the X operatorso that r(X) = r0 and Xx = X Then

Proof Suppose X = YX Then Y C Yx = X so Y C X If is anedge of X~ then certainly is in X mdash Yx If were in Yx Y then byLemma 132 X(YX f) = f but this means that is internally activewith respect to X = Yx contradicting euro X~ Thus is in Y andX~ QY

Suppose X- C Y C X If X = Y then we have X = Xx = YxNow if euro X Y then is internally active with respect to X and soX(Xf) = f Prom F C I w e have (by Lemma 133) X(Xf) e Yxthat is G Yx Since this is true for all in X Y it follows thatXY CYX and consequently X C Yx Finally from Definition 131and Y C X C Yx we deduce that Xx = Yx that is X = Yx D


We note the analogous result if X is in the image of the i operatorthen

Proposition 138 Let T be a spanning tree (that is T e V) andsuppose W pound C is such that Wx = T Then We = Te

Proof Suppose that the edge e is externally active with respect toT We shall show that the whole of cyc(T e) belongs to W whence itfollows that e is externally active with respect to W If there is an edgef ^ em cyc(T e) which is not in W then since (by Proposition 137)we have T~ C W C T must be internally active with respect to TNow 6 cyc(T e) implies that e euro cut(T) and the internally activeproperty of means that lt e This contradicts the externally activeproperty of e Hence cyc(T e) C W and e is externally active withrespect to W

Conversely if e is externally active with respect to W it follows im-mediately that e is externally active with respect to T bull

We now set up the main theorem using the portion A mdash C mdashgt Tgtof the operator diagram Define

Pii = X euro A | r(X) = r0 - t s(X) = j

ni = W 6 C | r(W) = r0 - t |W~| = j |

Of course the last line merely repeats Definition 136 We have threecorresponding two-variable polynomials

RT xy)=Yl Pa^V3gt P(rgt poundraquo) = pound wyVgt

T(r ltraquo) = ^ t y i V

where the modified rank polynomial R is related to the usual one (Defi-nition 101) by R(Txy) = a

Theorem 139 Let T be a connected graph with n vertices and let ltbe any ordering of ET Then the Tutte polynomial is related to the rankpolynomial as follows

T(rltx + ly + l) = R(rxy) = xn-lR(Tx-y)

Proof We shall use the intermediate polynomial P defined above andprove the equalities

TT ltx+ly + l) = P(T ltxy+l) = R(T xy)

which are equivalent to the following relationships among the coeffi-


cients

k x 7 i

For the first identity consider A C - V By Proposition 137 if T isin V then

T = WX if and only if T~ CWCT

Also by Proposition 138 the external activities of T and W are thesame Consequently for each one of the tkj spanning trees T with |X| =k and X^ = j there are () subgraphs W in C with r(W) = r0 -i andWe mdash j These subgraphs are obtained by removing from T any set ofi edges contained in the k internally active edges of T This proves thefirst identity

For the second identity we consider i A -+ C By the analogue ofProposition 137 for x if X is in C then

X^Y if and only if XCYCX+

Consequently for each one of the iru subgraphs X in C with r(X) =ro mdash i and Xe = I there are () subgraphs Y with r(Y) = r0 - i ands(Y) = j These subgraphs are obtained by adding to X any set of jedges contained in the I externally active edges of X This proves thesecond identity bull

Corollary 1310 The Tutte polynomial of a connected graph T isindependent of the ordering used in its definition

Proof This statement follows from Theorem 139 and the fact thatthe rank polynomial is independent of the ordering bull

The original proof of Theorem 139 by Tutte (1954) was inductive theproof given above is a simplification of the first constructive proof byCrapo (1969) In the light of the Corollary we can write T(Txy) forthe Tutte polynomial of T It should be noted that although eachcoefficient Uj is independent of the ordering the corresponding set ofspanning trees (having internal activity i and external activity j) doesdepend on the ordering

Additional Results

13a Tutte polynomials of cycles By listing the spanning trees of Cn

and calculating their internal and external activities we obtainT1 i i 2 i t n mdash 1


13b The Tutte matrix of Petersens graph (Biggs 1973b) The matrix(Uj) of coefficients of the Tutte polynomial for Petersens graph is

r 0 36 84 75 35 9 136 168 171 65 10120 240 105 15180 170 30170 70114 1256216

L l

13c The deletion-contraction property The following two propertiescompletely define the Tutte polynomial for connected graphs

(1) If e is an edge of the connected graph F which is neither a loopnor an isthmus then T(Txy) = T(r(e)cy) + T(r(e)xy)

(2) If Ajj is formed from a tree with i edges by adding j loopsT(AiJxy)=xiy

13d Recursive families (Biggs Damerell and Sands 1972) Using thedeletion-contraction property we can obtain a second-order recurrencefor the Tutte polynomials of the cycle graphs

T(Cn+2xy) - (x + l)T(Cn+1xy) + xT(Cnxy) = 0

Generally a family Ti of graphs is said to be a recursive family if thereis a linear recurrence of the form

T(Ti+p x y) + aiTYl+p^ xy) + + opT(r x y) = 0where the coefficients a j ap are polynomial functions of (x y) andare independent of Thus the cycle graphs form a recursive familywith p = 2 The families Lh Mh of ladders and Mobius laddersare recursive families with p mdash 6 they have the same recurrence whoseauxiliary equation is

(t - l)(t - x)(t2 -(x + y + 2)t + xy)t2 ~ (x2 + x + y+ l)t + x2y) = 0From this we can deduce the tree-numbers and the chromatic polyno-mials for these graphs (See also 9c)

13e Tutte polynomials of complete graphs Let T(X y a) and p(y a)be the exponential generating functions for the polynomials TKn x y)and y(2(y mdash l)~n respectively Then

r(xya) =x-1


13f Inversions of trees A labelled tree on n vertices is a spanning treeA of Kn with the vertex-set 12 n Let inv(A) denote the numberof edges ij of A for which i lt j and j is on the path in A from 1 toi Then we have

A

where the sum is over all labelled trees on n vertices

13g The commutative diagram If X C ET define

T = Xraquo U (Xx X) = Xx (X X)

Then Xxraquo = T = X^x (Crapo 1969)

13h Counting forests If we write T(T 11 + t) = poundamplt then fa isthe number of forests in V which have | ^ r | mdash i mdash 1 edges It followsthat T(T 12) is the total number of forests in T and T(T 11) is thetree-number of T

13i Planar graphs If T and T are dual planar graphs then there isa bijective correspondence between their spanning trees which switchesinternal and external activity It follows that tj = t^ and

TYxy)=TTyx)

13j The medial graph Let F be a connected graph which is embed-ded in the plane For each e euro E(T) choose an interior point m(e)on e The medial graph M(T) associated with the given embedding ofF has vertex-set m(e) | e euro E(T) and edge-set defined as followsFor each face of the embedded graph T there is a cycle with edgeseie2 bull ek bounding that face we create a corresponding sequencefî fi2 bull bullHk of edges of M(T) which (i) forms a cycle in M(T) withvertices m(ei)m(e2) m(efc) and (ii) is topologically identical withthe original cycle M(T) is a 4-regular graph and as such it has at leastone Eulerian partition that is a partition of its edge-set into cycles with-out repeated edges Let fk denote the number of Eulerian partitions ofM(G) into k cycles such that at any any vertex of M(G) the two cy-cles passing through that vertex do not cross in the obvious topologicalsense Las Vergnas (1978) proved that

fcgt0

See also Jaeger (1988) and Las Vergnas (1988)


13k Tutte polynomials for knots and links (Thistlethwaite 1987) Aknot or link L is usually represented by a diagram in the plane thediagram is said to be alternating if the crossings are alternately over andunder as we traverse each component Associated with an alternatingdiagram is a graph DL such that the Jones polynomial of L is given by

VLt) = -t)-KTDL-t-rl

where K is a number depending on LThis relationship leads to a simple proof of a conjecture made by Tait

in the 19th century the number of crossings in any alternating diagramof a given link is invariant provided there are no nugatory crossings

131 Intractability of calculating the Tutte polynomial A counting prob-lem is said to be P-hard if it has a certain technical property whichit is believed is equivalent to computational intractability Jaeger Ver-tigan and Welsh (1990) showed that computing T(Txy) is P-hardexcept for a few points and curves in the complex x y)-plane In par-ticular computing the Jones polynomial (13k) of an alternating link isP-hard

14

Chromatic polynomials and spanning trees

In this chapter we shall study the relationship between the Tutte poly-nomial and the chromatic polynomial of a connected graph The mainresult is as follows

Theorem 141 Let F be a connected graph with n vertices Thenn-l

C(T u) = (-l)-1^ ]T laquo(1 - )t=i

where poundJO is the number of spanning trees ofT which have internal activityi and external activity zero (with respect to any fixed ordering of EY)

Proof We have only to invoke some identities derived in earlier chap-ters The chromatic polynomial is related to the rank polynomial as inCorollary 105 and the rank polynomial is related to the Tutte polyno-mial as in Theorem 139 Thus we have

C(Tu) =unR(F-u-1-l)

The result follows from the definition of the Tutte polynomial bull

This theorem indicates a purely algebraic way of calculating chro-matic polynomials If we are given the incidence matrix of a graph Fthen the basic cycles and cuts associated with each spanning tree T ofF can be found by matrix operations as explained in Chapter 5 Promthis information we can compute the internal and external activities of

Chromatic polynomials and spanning trees 107

T using the results of Proposition 135 The method is impracticablefor hand calculation but it is well-adapted to automatic computation inview of the availability of sophisticated programs for carrying out ma-trix algebra Furthermore it is demonstrably better than the deletion-contraction method (see 14h)

Theorem 141 also has theoretical implications for the study of chro-matic polynomials and the remainder of this chapter is devoted to someof these consequences First we observe that if the chromatic polyno-mial is expressed in the reduced form

n-2

C(F u) = plusmnw(w mdash 1) VJ diW1 where w = 1 mdash ui=0

then the coefficients a are all non-negative In fact a is the numberU+ifi- It is convenient to use the reduced form to record chromaticpolynomials because the coefficients have fixed sign and are relativelysmall

Proposition 142 Let T be a connected graph and let (poundbdquobull) denotethe matrix of coefficients of its Tutte polynomial Then

Proof Suppose that the ordering of ET = ei e2 em is the nat-ural order of the subscripts If T is a spanning tree with internal activity1 and external activity 0 then ei must be an edge of T otherwise itwould be externally active Further e is not an edge of T otherwiseboth ei and t2 would be internally active Also e is in cyc(T e2) oth-erwise e2 would be externally active Consequently Tlaquo = (T e) U e isa spanning tree with internal activity 0 and external activity 1

Reversing the argument shows that T gt-+ T is a bijection and hencetio (the number of spanning trees T with |Ti = 1 and Teuro = 0) is equalto toi (the number of spanning trees T with T^ = 0 and |T| = 1)

bull

The number tw has appeared in the work of several authors for exam-ple Crapo (1967) and Essam (1971) We note that it is the coefficientao in the reduced form of the chromatic polynomial It is sufficientlyimportant to warrant a name

Definition 143 The chromatic invariant 9(T) of a connected graphF is the number of spanning trees of T which have internal activity 1and external activity 0


Theorem 141 provides another interpretation of 8(T) in terms of thechromatic polynomial of F Let C denote the derivative of C then asimple calculation shows that

When F is non-separable it has at least one spanning tree with internalactivity 1 and external activity 0 (14b) Thus for a non-separable graphwith an even number of vertices C is increasing at its zero u = 1 whereasif the graph has an odd number of vertices it is decreasing

The link with the chromatic polynomial can also be used to justifythe use of the name invariant for 0(F) Recall that two graphs are saidto be homeomorphic if they can both be obtained from the same graphby inserting extra vertices of degree two in its edges

Proposition 144 If I and F2 are homeomorphic connected graphswith at least two edges then

0(1^) = 0(T2)

Proof Let F be a graph which has at least three edges and a vertexof degree two Let e and be the edges incident with this vertex Thedeletion of either e or say e results in a graph r(e) in which theedge is attached at a cut-vertex to a graph To with at least one edgeHence C(rû) is of the form (u - 1)C(TO u) where C(F0 1) = 0 Thecontraction of e in F results in a graph homeomorphic with F We have

= (laquo-i)C(rou)-c(r(e)u)and on differentiating and putting u mdash 1 we find

C(rl) = -C(r ( e ) l )

Since F has one more vertex than F(e) it follows that

Now if two graphs are homeomorphic then they are related to somegraph by a sequence of operations like that by which F(e) was obtainedfrom F hence we have the result bull

It is worth remarking that both the proof and the result fail in the casewhere one of the graphs is K2 we have 6K2) = 1 whereas any pathgraph Pn (n gt 3) is homeomorphic with K2 but 0(Pn) = 0

We end this chapter with an application of Theorem 141 to the uni-modal conjecture of Read (1968) This is the conjecture that if

u) = un - Clun1 + + -l)n-lcn_lU


then for some number M in the range 1 lt M lt n mdash 1 we have

Cl lt C2 lt lt CM gt CM+l gt bull bull bull gt Cn-l-

There is strong numerical evidence to support this conjecture but aproof seems surprisingly elusive The following partial result was ob-tained by Heron (1972)

Proposition 145 Using the above notation for the chromatic poly-nomial of a connected graph F with n vertices we have

Ci-i lt ci for all i lt -n- 1)

Proof The result of Theorem 141 leads to the following expressionfor the coefficients of the chromatic polynomial

^ n-l-A ^ (n~l-l

U n-l-tj ^ -l JNow if iltn- 1) then i - I lt n - 1 - ) for all I gt 0 Hence bythe unimodal property of the binomial coefficients we have

Thus since each number poundn_i_j)o is a non-negative integer it followsthat Ci gt c_i for i lt | ( n mdash 1) as required

Additional Results

14a A product formula for 8 If F has a quasi-separation (Vi V2) withjVx 0 Vaj = t t h e n

This formula is particularly useful when t mdash 2

14b Graphs with a given value of 9 A connected graph F is separableif and only if 9(T) = 0 It is a series-parallel graph if and only if0(F) lt 1 (Brylawski 1971) One graph with 0 = 2 is 4 and it followsfrom Brylawskis result on series-parallel graphs that if F contains asubgraph homeomorphic to K4 then 0(F) gt 2 In order to show thatall values of 6 can occur we need only remark that for the wheel Wn wehave 0(Wn) = n mdash 2 Using the product formula 14a we can constructinfinitely many graphs with any given value of 9 by gluing any edge ofany series-parallel graph to any edge of the appropriate wheel


14c The chromatic invariants of dual graphs Let F and F be dualplanar connected graphs Then

0(r) = 0(r)For instance

0(Q3) = 0(222) = 11 O(Icosahedron) = 6(Dodecahedron) = 4412

14d Some explicit formulae For the complete graphs ifn the laddersLh and the Mobius ladders Mh we have

0(Kn) = (n - 2) (n gt 2)

0(Lh) = 2fe - ft - 1 (ft gt 3)

0(Mh) = 2h - ft (ft gt 2)

14e Tfte Zoiu polynomial Let C(F w) denote the number of nowhere-zero u-flows (see 4k) on a connected graph F with n vertices and medges Then

C(F u) = -l)mR(T - 1 -u) = ( - l )m-n + 1T(F 01 - laquo)

If F is planar and F is its dual then (Tutte 1954)

C(Tu)=uC(ru)

Thus the problem of finding the flow polynomial of a planar graph isequivalent to finding the chromatic polynomial of its dual For examplethe flow polynomial of a ladder graph can be derived from the chromaticpolynomial of its dual a double pyramid (9a)

The general relationship between the flow polynomial of a graph andan interaction model is discussed in Biggs (1977b Chapter 3)

14f The flow polynomials of Kzz and O3 From the rank matrix of^33 (Chapter 10) and the Tutte matrix of O3 (13b) we can obtain theflow polynomials for these (non-planar) graphs

C(K3s laquo) = ( laquo - l)(u - 2)(M2 - u + 10)

C(O3u) = u- l)(u - 2)(it - 3)(laquo - 4)(u2 -5u + 10)In both cases there is no graph whose chromatic polynomial is uC

14g Expansions of the flow polynomial Jaeger (1991) obtained an ex-pansion of the flow polynomial of a graph F of degree 3 imbedded inthe plane Define an even subgraph to be a subgraph (C) in which ev-ery vertex has even degree Since T has degree 3 this means that everycomponent of (C) is a cycle and so each component can be oriented in


one of two ways Associated with every oriented even subgraph (C) is aweight w(C) such that

C ( F (u + y - 1 ) 2 ) = ] pound ( V c

where p(C) is a rotation number depending on the relative orientationof the cycles of (C) with respect to the plane in which F is embedded

14h The superiority of the matrix method It follows from the result ofJaeger Vertigan and Welsh (131) that computing the chromatic polyno-mial is in general P-hard However there is some interest in compar-ing methods of computation even though they are all bad in theoreticalterms

The matrix method (call it Method A) described in our comments onTheorem 141 has been used only rarely (Biggs 1973b) However An-thony (1990) showed that it is more efficient than the method of deletionand contraction (Method B) even when that method incorporates rulesfor curtailing the computation Specifically the worst-case running timeof Method A for a graph with n vertices and m edges is of the order of(ntradei)n2m- ^ T_A(TI) and Ten) denote the worst-case running times ofthe respective methods for any sequence of graphs such that Fn has nvertices and the average degree A(n) mdashraquo oo as n mdashgt oo we have

log TB(n)log TAn)

bull oo as n mdashgt oo

PART THREE

Symmetry and regularity

15

Automorphisms of graphs

An automorphism of a (simple) graph F is a permutation n of VT whichhas the property that u v is an edge of F if and only if ir(u)Tr(v)is an edge of F The set of all automorphisms of F with the operationof composition is the automorphism group of F denoted by Aut(F)

Some basic properties of automorphisms are direct consequences ofthe definitions For example if two vertices x and y belong to the sameorbit that is if there is an automorphism a such that a(x) = y thenx and y have the same degree This and other similar results will betaken for granted in our exposition

We say that F is vertex-transitive if Aut(F) acts transitively on VTthat is if there is just one orbit This means that given any two verticesu and v there is an automorphism TT euro Aut(F) such that TT(U) = vThe action of Aut(F) on VT induces an action on ET by the rulenx y = n(x)7r(y) and we say that F is edge-transitive if this actionis transitive in other words if given any pair of edges there is an auto-morphism which transforms one into the other It is easy to constructgraphs which are vertex-transitive but not edge-transitive the laddergraph L3 is a simple example In the opposite direction we have thefollowing result

Proposition 151 If a connected graph is edge-transitive but notvertex-transitive then it is bipartiteProof Let x y be an edge of F and let X and Y denote the orbitscontaining x and y respectively under the action of Aut(F) on the ver-tices It follows from the definition of an orbit that X and Y are either

116 Symmetry and regularity

disjoint or identical Since F is connected every vertex z is in someedge zw and since F is edge-transitive z belongs to either X or YThus XUY = VT If X = Y = VT then F would be vertex-transitivecontrary to hypothesis consequently X n Y is empty Every edge of Fhas one end in X and one end in Y so F is bipartite D

The complete bipartite graph Ka^ with a ^ b is an obvious example ofa graph which is edge-transitive but not vertex-transitive In this casethe graph is not regular and it is not vertex-transitive for that reasonbecause it is clear that in a vertex-transitive graph each vertex must havethe same degree Examples of regular graphs which are edge-transitivebut not vertex-transitive are not quite so obvious but examples areknown (see 15c)

The next proposition establishes a link between the spectrum of agraph and its automorphism group We shall suppose that VT is theset viigt2 bull bull bull vn and that the rows and columns of the adjacencymatrix of F are labelled in the usual way A permutation -K of VT canbe represented by a permutation matrix P = (Pij) where Pij = 1 ifVi = IT(VJ) and Pij = 0 otherwise

Proposition 152 Let A be the adjacency matrix of a graph F andft a permutation of VT Then TT is an automorphism ofT if and only ifPA = AP where P is the permutation matrix representing n

Proof Let Vh mdash TT(VJ) and Vk = K(VJ) Then we have

(AP) hj = ZciMpij = ahkConsequently AP = PA if and only if Uj and VJ are adjacent wheneverVh and Vk are adjacent that is if and only if TT is an automorphism of

r D

A consequence of this result is that loosely speaking automorphismsproduce multiple eigenvectors corresponding to a given eigenvalue To beprecise suppose x is an eigenvector of A corresponding to the eigenvalueA Then we have

APx = PAx = PAx = APx

This means that Px is also an eigenvector of A corresponding to theeigenvalue A If x and Px are linearly independent we conclude thatA is not a simple eigenvalue The following results provide a completedescription of what happens when A is simple

Automorphisms of graphs 117

Lemma 153 Let A be an simple eigenvalue ofT and let x be a cor-responding eigenvector with real components If the permutation matrixP represents an automorphism of F then Px = plusmnx

Proof If A has multiplicity one x and Px are linearly dependentthat is Px = [jx for some complex number ft Since x and P are reali is real and since P = I for some natural number s gt 1 it followsthat p is an sth root of unity Consequently z = plusmn1 and the lemma isproved bull

Theorem 154 (Mowshowitz 1969 Petersdorf and Sachs 1969) all the eigenvalues of the graph F are simple every automorphism of F(apart from the identity) has order 2

Proof Suppose that every eigenvalue of F has multiplicity one Thenfor any permutation matrix P representing an automorphism of F andany eigenvector x we have P2x = x The space spanned by the eigen-vectors is the whole space of column vectors and so P2 = I D

Theorem 154 characterizes the group of a graph which has the maximumnumber n = |VT| of distinct eigenvalues every element of the groupis an involution and so the group is an elementary abelian 2-groupFor example the theta graph copy221 (K4 with one edge deleted) hasautomorphism group Z2 x Z2 The characteristic polynomial is

and so every eigenvalue is simple On the other hand if we know thata graph has an automorphism of order at least three then it must havea multiple eigenvalue In particular this means that the 2i numbersobtained in 3e as the eigenvalues of the Mobius ladder M21 cannot allbe distinct

The question of which groups can be the automorphism group of somegraph was answered by Frucht (1938) He showed that for every ab-stract finite group G there is a graph F whose automorphism group isisomorphic to G He also proved that the same result holds with F re-stricted to be a regular graph of degree 3 (Frucht 1949) Although thereare some gaps in the original proof satisfactory proofs of the result arenow available For an overview of this subject the reader is referred toBabai (1981) He describes how Fruchts work stimulated a great dealof research and how it has been extended by several authors to showthat the conclusion remains true even if we specify in advance that Fmust satisfy a number of graph-theoretical conditions


If we strengthen the question by asking whether every group of per-mutations of a set X is the automorphism group of some graph withvertex-set X then the answer is negative For example the cyclicpermutation-group of order 3 is not the automorphism group of anygraph with three vertices (It is of course a subgroup of the group ofK3) This tends to confirm our intuitive impression that there must besome constraints upon the possible symmetry of graphs One such con-straint is the following If F is a connected graph and d(u v) denotes thedistance in F between the vertices u and v then for any automorphisma we have

duv) = dau)av))

Thus there can be no automorphism which transforms a pair of verticesat distance r into a pair at distance s ^ r The following definitionframes conditions which are in a sense partially converse to this result

Definition 155 Let F be a graph with automorphism group Aut(F)We say that F is symmetric if for all vertices uvxy of F such that uand v are adjacent and x and y are adjacent there is an automorphisma in Aut(F) for which a(u) = x and a(v) = y We say that F is distance-transitive if for all vertices uvxy of F such that d(u v) mdash d(x y) thereis an automorphism a in Aut(F) satisfying a(u) = x and a(v) mdash y

It is clear that we have a hierarchy of conditions

distance-transitive =gtbull symmetric =gt vertex-transitive

In the following chapters we shall investigate these conditions in turnbeginning with the weakest one

Additional Results

15a How large can an automorphism group be For any value of nthe automorphism group of the complete graph Kn contains all the npermutations of its n vertices it is the symmetric group Sn- Any othergraph on n vertices has an automorphism group which is a subgroup ofSn Since the complete graph is the only connected graph in which eachpair of distinct vertices is at the same distance it is the only connectedgraph for which the automorphism group can act doubly-transitively onthe vertex-set


15b How small can an automorphism group be Except for very smallvalues of n it is easy to construct a graph with n vertices which has thetrivial automorphism group containing only the identity permutationFor n gt 7 the tree with n vertices shown in Figure 7 is an example

Figure 7 a tree with no non-trivial automorphisms

In fact almost all graphs have the trivial automorphism group The fullstory is described by Bollobas (1985 Chapter 9)

15c A regular graph which is edge-transitive but not vertex-transitiveConsider a cube divided into 27 equal cubes in the manner of Rubikscube and let us say that a row is a set of three cubes in a row parallelto a side of the big cube Define a graph whose vertices are the 27 cubesand the 27 rows a cube-vertex being adjacent to the three row-verticesto which it belongs This example of a regular edge-transitive graphwhich is not vertex-transitive is the first of a family of examples due toBouwer (1972)

15d The automorphism groups of trees (Jordan 1869) Let T be a finitetree Then either (i) T has a vertex v known as the centroid which isfixed by every automorphism of T or (ii) T has an edge x t knownas the bicentroid which is fixed (setwise) by every automorphism of T

15e The graphs Pht) The generalized Petersen graph P(ftpound) isa 3-regular graph with 2h vertices xo Xi XH-U Vo 2i bull bull bull Vh-i andedges xi ylt xi xi+i yi y+th f o r a11 e 01 i raquo 1 wherethe subscripts are reduced modulo h For example P(h 1) is the laddergraph Lh and P(52) is Petersens graph Frucht Graver and Watkins(1971) showed that(a) P(ft t) is vertex-transitive if and only if t2 = plusmn1 (mod ft) or (ft t) =(102)(b) P(ht) is symmetric if and only if (ft t) is one of (41) (52) (83)(102) (103) (125) (245)


Case-by-case checking of the latter result shows that P(h t) is distance-transitive if and only if (ht) is one of (41) (52) (103)

15f The connection between Aut(r) and Aut(pound(F)) (Whitney 1932c)The automorphism groups of F and its line graph L(F) are not necessar-ily isomorphic for example Ki = L(K2) so in this case the first groupis trivial but the second is not However this is a rare phenomenonThere is a group homomorphism 0 Aut(r) mdashgt Aut(Z(F)) defined by

9g)u v = 0uOv) where g euro Aut(r) u v euro ET)

and we have (i) 6 is a monomorphism provided F ^ K2 (ii) 0 is anepimorphism provided F is not K4 K4 with one edge deleted or K4with two adjacent edges deleted

15g Homogeneous graphs A graph F is said to be weakly homogeneousif whenever two subsets U U2 of VF are such that (Ui) and (fjj) areisomorphic then at least one isomorphism between them extends to anautomorphism of F The complete list of weakly homogeneous graphs isas follows

(a) The cycle graph C5(b) The disjoint union of t gt 1 copies of the complete graph Kn(c) The complete multipartite graphs iiTSjSgts with t gt 2 parts of

equal size s(d) The line graph L(K3gt3)

A graph is homogeneous if whenever two subsets U U2 of VT are suchthat (Ui) and (U2) are isomorphic then every isomorphism betweenthem extends to an automorphism of F It is obvious that a homo-geneous graph is weakly homogeneous and somewhat surprisingly theconverse is also true This result has a contorted history The 1974version of this book caused some confusion by attributing to Sheehanthe classification of weakly homogeneous graphs given above In factSheehan (1974) obtained the classification of homogeneous graphs Gar-diner observed the error in the book and then (1976) obtained the samelist for the weakly homogeneous case by an independent method Fi-nally Ronse (1978) showed directly that a weakly homogeneous graph ishomogeneous

15h Graphs which are transitive on vertices and edges Let F be a graphfor which Aut(F) acts transitively on both vertices and edges Then Fis a regular graph and if its degree is odd it is symmetric (Tutte 1966)If its degree is even the conclusion may be false as was first shown


by Bouwer (1970) Holt (1981) gave an example of a 4-regular graphwith 27 vertices which is vertex-transitive and edge-transitive but notsymmetric and Alspach Marusic and Nowitz (1993) showed that Holtsexample is the smallest possible

15i Graphs with a given group (Izbicki 1960) Let an abstract finitegroup G and natural numbers r and s satisfying r gt 3 2 lt s lt r begiven Then there are infinitely many graphs F with the properties

(a) Aut(F) is isomorphic to G(b) F is regular of degree r(c) the chromatic number of F is s

16

Vertex-transitive graphs

In this chapter we study graphs F for which the automorphism group actstransitively on VT As we have already noted in the previous chaptervertex-transitivity implies that every vertex has the same degree so Fis a regular graph

We shall use the following standard results on transitive permutationgroups Let G mdash Aut(F) and let Gv denote the stabilizer subgroup forthe vertex v that is the subgroup of G containing those automorphismswhich fix v In the vertex-transitive case all stabilizer subgroups Gv (v 6VF) are conjugate in G and consequently isomorphic The index of Gv

in G is given by the equation

G Gv = GGV = |VT|

If each stabilizer Gv is the identity group then every element of G(except the identity) does not fix any vertex and we say that G actsregularly on VT In this case the order of G is equal to the number ofvertices

There is a standard construction due originally to Cayley (1878)which enables us to construct many but not all vertex-transitive graphsWe shall give a streamlined version which has proved to be well-adaptedto the needs of algebraic graph theory Let G be any abstract finitegroup with identity 1 and suppose ft is a set of generators for G withthe properties

(i) x g Q =gt re1 G fi (ii) 1 pound fi

Vertex-transitive graphs 123

Definition 161 The Cayley graph T = F(G pound2) is the simple graphwhose vertex-set and edge-set are defined as follows

VT = G EY = ghg-lheurott

Simple verifications show that ET is well-defined and that T(G pound2) isa connected graph For example if G is the symmetric group S3 andpound2 = (12) (23) (13) then the Cayley graph TGQ) is isomorphic to33 (Figure 8)

l (12)

(123)

(132) (23)

Figure 8 K3t3 as a Cayley graph for 53

Proposition 162 (1) The Cayley graph T(G pound2) is vertex-transitive(2) Suppose that ir is an automorphism of the group G such that TT(pound2) =pound2 Then n regarded as a permutation of the vertices ofT(GCl) is agraph automorphism fixing the vertex 1

Proof (1) For each g in G we may define a permutation ~g of VT = Gby the rule g(h) = gh (h euro G) This permutation is an automorphismof T for

hkGET=gth~lkefl

= (gh)~lgk euro Q

The set of all g (g euro G) constitutes a group G (isomorphic with G)which is a subgroup of the full group of automorphisms of F(G pound2) andacts transitively on the vertices

(2) Since IT is a group automorphism it must fix the vertex 1 Fur-thermore n is a graph automorphism since

h jfc euro J5r =gt i-1fc euro pound2 =gt Ttih^k) euro pound2

a

The second part of this proposition implies that the automorphismgroup of a Cayley graph F(G 0) will often be strictly larger than G


In the example illustrated in Figure 8 every group automorphism ofS3 fixes fl setwise and so it follows that the stabilizer of the vertex1 has order at least 6 In fact the order of the stabilizer is 12 and|Aut(K33)| = 72

Not every vertex-transitive graph is a Cayley graph for example Pe-tersens graph O3 is not a Cayley graph This statement can be checkedby noting that there are only two groups of order 10 and they havefew generating sets of size three satisfying the conditions in Definition161 An exhaustive check of all the possibilities confirms that Petersensgraph does not arise as a Cayley graph in this way

We begin our study of the hierarchy of symmetry conditions with thecase when Aut(F) acts regularly on V(T)

Lemma 163 Let F be a connected graph Then a subgroup H ofAut(F) acts regularly on the vertices if and only if F is isomorphic to aCayley graph T(H Q) for some set Q which generates H

Proof Suppose VT = v v2 vn and H is a subgroup of Aut(F)acting regularly on VT Then for 1 lt i lt n there is a unique hi euro Hsuch that hi(vi) = igtj Let

Cl = hi euro H I Vi is adjacent to vi in F

Simple checks show that Q satisfies the two conditions required by Def-inition 161 and that the bijection Vi lt-gt hi is a graph isomorphism of Fwith T(HQ) Conversely if F = F(Q) then the groupjf defined inthe proof of Proposition 162 acts regularly on VT and H laquo H bull

Lemma 163 shows that if Aut(F) itself acts regularly on VT then Fis a Cayley graph F(Aut(F) O)

Definition 164 A finite abstract group G admits a graphical regularrepresentation or GRR if there is a graph F such that G is isomorphicwith Aut(F) and Aut(F) acts regularly on VT

The question of which abstract groups admit a GRR was answeredcompletely in the late 1970s (see 16g) It turns out that the secondpart of Proposition 162 is essentially the only obstacle to there being aGRR for G In other words a group G has no GRR if and only if everygenerating set Cl for G which satisfies conditions (i) and (ii) is such thatthere is an automorphism of G fixing Q setwise

As an example of the ideas involved we show that the group S3 admitsno graphical regular representation If there were a suitable graph Fthen it would be a Cayley graph F(53 f2) Now it is easy to check by an


exhaustive search that for any generating set 0 satisfying conditions (i)and (ii) on p 122 there is some automorphism of 53 fixing Q setwiseThus by part (2) of Proposition 162 the automorphism group of aCayley graph T(S$ 0) is strictly larger than S3

In the case of transitive abelian groups precise information is providedby the next proposition

Proposition 165 Let F be a vertex-transitive graph whose automor-phism group G = Aut(F) is abelian Then G acts regularly on VT andG is an elementary abelian 2-group

Proof If g and h are elements of the abelian group G and g fixes vthen gh(v) = hg(v) = h(v) so that g fixes h(v) also If G is transitiveevery vertex is of the form h(v) for some h in G so g fixes every vertexThat is g = 1

Thus G acts regularly on VT and so by Lemma 163 F is a Cayleygraph F(G fi) Now since G is Abelian the function g raquo-gt g~x is anautomorphism of G and it fixes Q setwise If this automorphism werenon-trivial then part (2) of Proposition 162 would imply that G is notregular Thus g = g1 for all g euro G and every element of G has order2 bull

We now turn to a discussion of some simple spectral properties ofvertex-transitive graphs A vertex-transitive graph F is necessarily aregular graph and so its spectrum has the properties which are statedin Proposition 31 In particular if F is connected and regular of degreek then k is a simple eigenvalue of F It turns out that we can use thevertex-transitivity property to characterize the simple eigenvalues of F

Proposition 166 (Petersdorf and Sachs 1969) Let F be a vertex-transitive graph which has degree k and let X be a simple eigenvalue ofF |VT| is odd then X = k If VT is even then X is one of theintegers 2a mdash k (0 lt a lt k)

Proof Let x be a real eigenvector corresponding to the simple eigen-value A and let P be a permutation matrix representing an automor-phism K of F If ir(vi) = Vj then by Lemma 153

Xi = (PX)- = plusmnXj

Since F is vertex-transitive we deduce that all the entries of x havethe same absolute value Now since u = [1 1 1] is an eigenvectorcorresponding to the eigenvalue k if A ^ k we must have ux = 0 thatis Yl xi = 0- This is impossible for an odd number of summands of equalabsolute value and so our first statement is proved


If F has an even number of vertices choose a vertex Vi of T and supposethat of the vertices Vj adjacent to vu a number a have Xj = x whilek mdash a have Xj = mdashX Since (Ax)i = Xxi it follows that Y^ xj = ^xigtwhere the sum is taken over vertices adjacent to laquo Thus

axi mdash (k mdash a)xi = Xxi

whence A = 2a mdash k bull

For example the only numbers which can be simple eigenvalues of a 3-regular vertex-transitive graph are 31 mdash1 mdash3 This statement is false ifwe assume merely that the graph is regular of degree 3 many examplescan be found in [CvDS pp 292-305]

If we strengthen the assumptions by postulating that T is symmetricthen the simple eigenvalues are restricted still further

Proposition 167 Let T be a symmetric graph of degree k and let Abe a simple eigenvalue ofT Then A = plusmnk

Proof We continue to use the notation of the previous proof Let Vjand vi be any two vertices adjacent to laquoraquo then there is an automorphism7T of F such that n(vi) = Vi and K(VJ) = uj If P is the permutationmatrix representing n then n(vi) = Vi implies that Px = x and soXj = x Thus a mdash 0 or k and A = plusmnk

We remark that the eigenvalue -k occurs and is necessarily simpleif and only if T is bipartite

Additional Results

16a Circulant graphs A circulant graph is vertex-transitive and aconnected circulant graph is a Cayley graph F(Zn 0) for a cyclic groupZn Adam (1967) conjectured that if two such graphs r(Znfl) andF(Zn0 ) are isomorphic then O = zQ for some invertible element zin Z n Elspas and Turner (1970) showed that the conjecture is true ifn is a prime or if the graphs have only simple eigenvalues but falsein general Parsons (1980) showed that it is true if both graphs havevertex-neighbourhoods isomorphic to the cycle C^

16b The ladder graphs as Cayley graphs The dihedral group Z2n oforder 2n is defined by the presentation

The Cayley graph of poundgt2n with respect to the generating set xx~~lyis the ladder graph Ln


16c Cayley graphs for the tetrahedral and icosahedral groups The al-ternating group An is the subgroup of index two in Sn containing allthe even permutations The groups A4 and A5 are sometimes known asthe tetrahedral and icosahedral groups because they are isomorphic withgroups of rotations of the respective polyhedra Both groups can berepresented by planar Cayley graphs A Cayley graph for At is shownin Figure 9

Figure 9 a Cayley graph for A

A Cayley graph for A5 is the skeleton of the famous carbon-60 structurealso known as buckminsterfullerene or the buckie-ball or the soccerball

16d The stabilizer of a vertex-neighbourhood Suppose that F is avertex-transitive graph with G = Aut(F) For any vertex v of F de-fine

Lv = g e Gv I g fixes each vertex adjacent to v

Then Lv is a normal subgroup of Gv More explicitly there is a homo-morphism from Gv into the group of all permutations of the neighboursof v with kernel Lv It follows from this that GV Lv lt k where k isthe degree

16e The order of the vertex-stabilizer Let Hn be the graph formed bylinking together n units of the form shown in Figure 10 so that theyform a complete circuit Then the graphs Hn are vertex-transitive andthe order of the vertex-stabilizer (2trade) is not bounded in terms of thedegreeOn the other hand in a symmetric graph the order of the vertex-stabilizer is bounded in terms of the degree See 17g


Figure 10 the vertex-stabilizer is not bounded

16f Coset graphs Let G be an abstract finite group H a subgroup of Gand fl a subset of GH such that 1 $ O fi1 = fi and if UQ generatesG The simplest way of denning a (general) graph whose vertices are theright cosets of H in G is to make Hg and Hgi adjacent whenever g29X

is in fi The graph so constructed is connected and vertex-transitiveThere are other ways of defining a graph whose vertices are cosets

and some of them result in a symmetric graph Examples and furtherreferences may be found in a paper by Conder and Lorirfier (1989)

16g Graphical regular representations Hetzel (1976) proved that theonly solvable groups which have no GRR are(a) abelian groups of exponent greater than 2(b) generalized dicyclic groups(c) thirteen exceptional groups such as the elementary abelian groupsZ| Z2Z| the dihedral groups DeDsDi0 and the alternating groupAThis work subsumed earlier results by several other authors Godsil(1981) showed that every non-solvable group has a GRR so the listgiven above is the complete list of groups which have no GRR

16h The eigenvalues of a Cayley graph (Babai 1979) Let T(G Q) bea Cayley graph and suppose that the irreducible characters of G areXij X2gt bull bull bull Xc with degrees m n2 nc respectively Then the eigen-values of F fall into families (A)j 1 lt i lt c 1 lt j lt n such thateach (Ai)j contains rii eigenvalues all with a common value Ajj (Notethat the total number of eigenvalues is thus ^Znf which is the correctnumber G) The sum of the tth powers of the Ay corresponding to agiven character satisfies

where the sum on the right-hand side is taken over all products of telements of 0


16i The Paley graphs Denote the additive group of the field GF(q)by Gq and let fi be the set of non-zero squares in GF(q) If q = 1 (mod4) then fi generates Gq and satisfies the conditions at the foot of p 122(remembering that the identity of Gq is the zero element of the field)The Paley graph P(q) is the Cayley graph F(G9O) These graphs arestrongly regular and self-complementary If q is the rth power of a primethe order of Aut(P(g)) is rq(q - l)2

16j Graphs with a specified vertex-neighbourhood A graph is said to belocally K if for each vertex v the subgraph induced by the neighbours ofv is isomorphic to K For example the graphs which are locally Petersenwere determined by Hall (1980) there are just three of them having21 63 and 65 vertices Many other papers on this topic are listed byBlokhuis and Brouwer (1992)

16k Generators for the automorphism group Let V be a connectedvertex-transitive graph and let Gv denote the stabilizer of the vertexv If h is any automorphism of T for which d(vh(v)) = 1 and T issymmetric then h and Gv generate Aut(F)

17

Symmetric graphs

The condition of vertex-transitivity is not a very powerful one as isdemonstrated by the fact that we can construct at least one vertex-transitive graph from each finite group by means of the Cayley graphconstruction A vertex-transitive graph is symmetric if and only if eachvertex-stabilizer Gv acts transitively on the set of vertices adjacent to vFor example there are just two distinct 3-regular graphs with 6 verticesone is 1(33 and the other is the ladder L3 Both these graphs are vertex-transitive and - 33 is symmetric but L3 is not because there are twokinds of edges at each vertex

Although the property of being symmetric is apparently only slightlystronger than vertex-transitivity symmetric graphs do have distinctiveproperties which are not shared by all vertex-transitive graphs This wasfirst demonstrated by Tutte (1947a) in the case of 3-regular graphs Morerecently his results have been extended to graphs of higher degree and ithas become apparent that the results are closely related to fundamentalclassification theorems in group theory (See 17a 17f 17g)

We begin by defining a t-arc [a] in a graph F to be a sequence(ao ci at) of t+ 1 vertices of F with the properties that a_i ais in ET for 1 lt i lt t and on- bull=pound on+ for 1 lt i lt t mdash 1 A t-arc is notquite the same thing as the sequence of vertices underlying a path oflength t because it is convenient to allow repeated vertices We regarda single vertex u a s a 0-arc [v] If = (0ofii 3s) is an s-arc inF then we write [a(3 for the sequence (ao bull bull at 3o bull bull bull Ps) provided

Symmetric graphs 131

that this is a (t + s + l)-arc that is provided at is adjacent to 30 andQf-i yen 0o at TÂ-

Definition 171 A graph F is t-transitive (t gt 1) if its automorphismgroup is transitive on the set of i-arcs in F but not transitive on the setof (t + l)-arcs in F

There is little risk of confusion with the concept of multiple transitivityused in the general theory of permutation groups since (as was noted in15a) the only graphs which are multiply transitive in that sense are thecomplete graphs We observe that the automorphism group is transitiveon 1-arcs if and only if F is symmetric (since a 1-arc is just a pair ofadjacent vertices) Consequently any symmetric graph is i-transitivefor some t gt 1

The only connected graph of degree one is K2 and this graph is 1-transitive The only connected graphs of degree two are the cycle graphsCn n gt 3) and these are anomalous in that they are transitive on t-arcs for all t gt 1 Prom now on we shall usually assume that the graphsunder consideration are connected and regular of degree not less thanthree For such graphs we have the following elementary inequality

Proposition 172 Let F be a t-transitive graph whose degree is atleast three and whose girth is g Then

Proof F contains a cycle of length g which is in particular a g-axcBecause the degree is at least three we can alter one edge of this g-avcto obtain a lt-arc whose ends do not coincide Clearly no automorphismof F can take a g-axc of the first kind to a g-axc of the second kind soit follows that t lt g

Pg-t=O0-

[a]

Figure 11 illustrating the proof of Proposition 172

Consequently if we select a cycle of length g in F then there is a t-arc


[a] without repeated vertices contained in it Let [] be the (g - t)-arcbeginning at at and ending at Qo which completes the cycle of lengthg Also let v be a vertex adjacent to at~i but which is not at-2 or atthis situation is depicted in Figure 11 Since F is ^-transitive there isan automorphism taking the t-arc [a] to the t-axc (aoa Qt-iv)This automorphism must take the (g mdash t + l)-arc [at-i3] to another(g mdash t + l)-arc [at-1-7] where 70 = v and ^g-t mdash ampo- The two arcsott-i-fi] and [at-17] may overlap but they define a cycle of length atmost 2(g-t + 1) Hence g lt 2(g - t + 1) that is ggt2t-2 bull

Definition 173 Let [a] and [] be any two s-arcs in a graph F Wesay that [0 is a successor of [a] if 3lt = aj+i ( 0 lt i lt s mdash 1)

It is helpful to think of the operation of taking a successor of [a] interms of shunting [a] through one step in F Suppose we ask whetherrepeated shunting will transform a given s-arc into any other If thereare vertices of degree one in F then our shunting might be halted ina siding while if all vertices have degree two we cannot reverse thedirection of our train However if each vertex of F has degree notless than three and F is connected then our intuition is correct andthe shunting procedure always works The proof of this requires carefulexamination of several cases and may be found in Tuttes book (Tutte1966 pp 56-58) Formally the result is as follows

Lemma 174 Let F be a connected graph in which the degree of eachvertex is at least three If s gt 1 and [a] [] are any two s-arcs in Fthen there is a finite sequence [ a^] (1 lt i lt I) of s-arcs in F suchthat [a1] = [a] [a()] = [] and [a(i+1gt] is a successor of [aW] for

We can now state and prove a convenient test for ^-transitivity LetF be a connected graph in which the degree of each vertex is at leastthree and let [a] be a i-arc in F

Figure 12 a i-arc and its successors


Suppose (as in Figure 12) that the vertices adjacent to at are at-i andv(1)vlt2gtv( i ) and let |W] denote the t-arc ( a i a 2 at v^) for1 lti lt I so that each [0^] is a successor of [a]

Theorem 175 Let T be a connected k-regular graph with I = k mdash 1 gt3 and let [a] be a t-arc in T Then Aut(F) is transitive on t-arcs ifand only if it contains automorphisms gig-gi such that gi[a] =[3(i)] (1 lt i lt I)

Proof The condition is clearly satisfied if Aut(r) is transitive on t-arcs Conversely suppose the relevant automorphisms ltilt2gt bull bull bull gi canbe found then they generate a subgroup H = (gi g2 gi) of Aut(F)and we shall show that H is transitive on t-arcs

Let [8] be a i-arc in the orbit of [a] under H thus [8] mdash ft [a] for someh e H If [4gt] is any successor of [8] then h~x[ltjgt] is a successor of [a]and so [ltjgt] = hgt[a] for some i euro 1 2 That is [ltjgt] is also in theorbit of [a] under H Now Lemma 174 tells us that all t-arcs can beobtained from [a] by repeatedly taking successors and so all t-arcs arein the orbit of [a] under if bull

As an example consider Petersens graph O3 whose vertices arethe unordered pairs from the set 12345 with disjoint pairs be-ing adjacent The automorphism group is the group of all permuta-tions of 12345 acting in the obvious way on the vertices Sincethe girth of O3 is 5 Proposition 172 tells us that the graph is atmost 3-transitive The 3-arc [a] mdash (12341523) has two successors[3ltx)] = (34152314) and [3lt2gt] = (34152345) The automorphism(13)(245) takes [a] to [(1)] and the automorphism (13524) takes [a] to

^ hence O3 is 3-transitive

In addition to its usefulness as a test for t-transitivity Theorem 175also provides a starting point for theoretical investigations into the struc-ture of t-transitive graphs Suppose that T is a connected t-transitivegraph (t gt 1) which is regular of degree fc gt 3 and let [a] be a givent-arc in F

Definition 176 The stabilizer sequence of [a] is the sequence

Aut(r) = G gt Ft gt Ft-t gt gtFigtF0

of subgroups of Aut(r) where F (0 lt i lt t) is defined to be thepointwise stabilizer of the set QO a i ctt-i-

In the case of Petersens graph with respect to the 3-arc (12341523)the group FQ is trivial Fi is the group of order 2 generated by (34) F2


is the group of order 4 generated by (34) and (12) and F3 is the groupof order 12 generated by (34) (12) and (345)

In general since G is transitive on s-arcs (1 lt s lt t) all stabilizersequences of pound~arcs are conjugate in G and consequently we shall oftenomit explicit reference to [a]

The order of each group occuring in the stabilizer sequence is de-termined by the order of FQ as follows Since Ft is the stabilizer ofthe single vertex ao in the vertex-transitive group G it follows thatG Ft = n = |VT| Since G is transitive on 1-arcs Ft acts transitivelyon the k vertices adjacent to ao and Ft- is the stabilizer of the vertexQi in this action consequently Ft Ft-i = k Since G is transitiveon s-arcs (2 lt s lt t) the group Ft-s+i acts transitively on the k mdash 1vertices adjacent to a s_i (other than as_2) and F t_ s is the stabilizerof the vertex as in this action consequently |Flt_S+1 F t_ s | = k mdash 1 for2 lt s lt t

Thus we have

G=nk(k-l)t-1F0This confirms our earlier observations about Petersens graph where wehave t = 3 and |F0 | = 1 so that |Fi| = 2 |F2 | = 4 |F3 | = 12 andG = 120

We shall now explain how the properties of the stabilizer sequence canbe conveniently discussed in terms of the set g gi gi of = k mdash 1automorphisms whose existence is guaranteed by Theorem 175 Definean increasing sequence of subsets of G = Aut(F) denoted by 1 = YoCYi C y2 C as follows

Yt = g-gl | ab euro 12 1 and 1 lt j lt i

Proposition 177 (1) If 1 lt i lt t then Yi is a subset of Fit but nota subset oFj_i (2) IfOltilt t then Fj is the subgroup of G generatedby Yi and FQ

Proof (1) For 1 lt a lt I we have g^ltUj) = Q-j+r provided that bothj and j + r lie between 0 and t Also ltpound~-+1(Qj) mdash v^- I followsthat gZsectb fixes ao ot ctt-i for all j lt i and so Yi C Fraquo If it weretrue that Yi C Fj_i then g^g would fix at-i+igt but this means thatgl

a(at-i+i) mdash glb(at-i+i) that is v^ = v^bh Since this is false for a^b

we have 1 Fj_i(2) Suppose euro Fi and [a] = (a o a i bull - bull a t - i 7 i bull bull bull 7raquo)- Pick any


gb since 7 is adjacent to at-i glili) is adjacent to gi(at-i) - at andso flî(7i) = laquo(o) for some a euro 12 Then

9al9lfa = (aoaiat-i+162-6i) say

By applying the same method with i replaced by i - 1 we can findan automorphism gcl~X) g1^1 which belongs to both Yî and yraquo andtakes 62 to at-i+2 while fixing ao a i a t _ i + i Continuing in thisway we construct g in Yt such that gf[a] mdash [a] that is gf is in FoConsequently is in the group generated by Yi and Fo Converselyboth Yi and Fo are contained in Fi so we have the result bull

All members of the sets YQ Y Yt fix the vertex a0 and so belongto Ft the stabilizer of ao further we have shown that Ft is generatedby Yt and Fo In the case of Yj+i we note that this set contains someautomorphisms not fixing ao and we may ask whether Yj+i and Fosuffice to generate the entire automorphism group G The followingproposition shows that the answer is ye s unless the graph is bipartiteThe reason why bipartite graphs are exceptional in this respect is thatif F is a symmetric bipartite graph in which VT is partitioned intotwo colour-classes V and V2 then the automorphisms which fix V andV2 setwise form a subgroup of index two in Aut(F) We say that thissubgroup preserves the bipartition

Proposition 178 Let T be a t-transitive graph with t gt 2 and girthgreater than 3 Let G denote the subgroup of G = Aut(F) generated byYt+i and Fo Then either (1) G = G or (2) F is bipartite GG = 2and G is the subgroup of G preserving the bipartition

Proof Let u be any vertex of F such that d(u ao) = 2 we showfirst that there is some g in G taking ao to u Since the girth ofF is greater than 3 the vertices w(a) = a+1(o) and u(6) = gpound+1(a0)satisfy dvâv^) = 2 Consequently the distance between ao andga^t+1^gl+1(cto) is also 2 Now G contains Ft (since the latter is gen-erated by Yt which is a subset of Yt+i and Fo) and Ft is transitiveon the 2-arcs which begin at a0 (since t gt 2) Thus G containsan automorphism fixing ao and taking ga + 9b+1(ao) to u andg = fga 9b+1 takes ao to u

Let U denote the orbit of ao under the action of G U contains allvertices whose distance from a0 is two and consequently all verticeswhose distance from a0 is even If U = VT then G is transitive onVT and since it contains Ft the stabilizer of the vertex ao in (G) isFt Thus |G| = |VT||Ft| = G and so G = G If U =pound V then U


consists precisely of those vertices whose distance from a^ is even andF is bipartite with colour-classes U and VT U Since G fixes themsetwise G is the subgroup of G preserving the bipartition bull

We remark that the only connected graphs of girth three whose auto-morphism group is transitive on 2-arcs are the complete graphs Thusthe girth constraint in Proposition 178 is not very restrictive

In the next chapter we shall specialize the results of Propositions177 and 178 to 3-regular graphs our results will lead to very preciseinformation about the stabilizer sequence

Additional Results

17a The significance of the condition t gt 2 In 16d we observed thatthe vertex-stabilizer Gv has a normal subgroup Lv such that GvLv isa group of permutations of the vertices adjacent to D In the case ofa symmetric graph with t gt 2 this group of permutations is doubly-transitive Since all doubly-transitive permutation groups are knownthis observation links the problem of classifying symmetric graphs withthe classification theorems of group theory See also 17f and 17g

17b The stabilizer of an edge-neighbourhood Suppose that F is a sym-metric graph of degree k with G mdash Aut(F) For any edge vw of Fdefine Gvw mdash GvnGw Lvw = LVCLW where Lv and Lw are the stabi-lizers of the respective vertex-neighbourhoods as defined in 16d Thenwe have the following subgroup relationships among these groups(a) Lv is a normal subgroup of Gv and Gvw(b) Lvw is a normal subgroup of Lv and GvwIt follows from standard theorems of group theory that

Jjy LtyLiU

jLjyyj J-JW

and LVLWLW is a normal subgroup of GvwLw The last group is agroup of permutations of the neighbours of w fixing v Thus we haveLV Lvw lt (k - 1) and

GV lt k(k - 1)LVW

17c The full automorphism group of Kn^n It is clear that the graphKn^n has at least 2(n)2 automorphisms Simple arguments suffice toshow that there are no others but for the sake of example we can use17b In this case the neighbourhood of an edge is the whole graph soLvw = 1 It follows that

G lt 2nGv lt 2nn (n - 1) = 2 (n)2


17d The automorphism group of Ok A more substantial applicationof 17b shows that the symmetric group S21C-1 is the full automorphismgroup of Ok- When k gt 3 every 3-arc in Ok determines a unique 6-cycleand it follows from this that if g e Lvw then g euro Lwx for all vertices xadjacent to w Hence Lvw = 1 and the order of the full automorphismgroup is at most

An alternative proof using the Erdos-Ko-Rado theorem may be foundin Biggs (1979)

17e The stabilizer sequence for odd graphs The odd graphs Ok are

3-transitive for all k gt 3 The stabilizer sequence is

G = S2k-u F3 = SkxSk-i F2 = Sk-i x Sk-i

Fi = S_i x Sk-2 Fo = Sk-2 x Sk-2-

17f Lvw is a p-group (Gardiner 1973) For any t-transitive graph witht gt 2 the edge-neighbourhood stabilizer Lvw is a p-group for some primep If t gt 4 and the degree is p + 1 it follows that the order of a vertex-stabilizer Gv is (p + l)pt~1m where t = 45 or 7 and m is a divisor ofp-lf

17g There are no 8-transitive graphs Weiss (1983) extended the resultsof Gardiner and others and using the classification theorems of grouptheory he showed that there are no finite graphs (apart from the cycles)for which a group of automorphisms can act transitively on the pound-arcsfor t gt 8 7-transitive graphs do exist the smallest is a 4-regular graphwith 728 vertices [BCN p 222]

17h Symmetric cycles A cycle with vertices VQ V vi~ in a graphF is symmetric if there is an automorphism g of F such that g(vi) = Vi+iwhere the subscripts are taken modulo JH Conway observed thatin a symmetric graph of degree k the symmetric cycles fall into fc mdash 1equivalence classes under the action of the automorphism group Thedetails may be found in Biggs (1981a) For example the two classesin Petersens graph contain 5-cycles and 6-cycles and in general theclasses in Ok have lengths 610 4fc mdash 6 and 2k mdash 1

18

Symmetric graphs of degree three

In this chapter we shall use the traditional term cubic graph to denotea simple connected graph which is regular of degree three As we shallsee the theory of symmetric cubic graphs is full of strange delights

Suppose that F is a t-transitive graph so that by definition Aut(F)is transitive on the f-arcs of F but not transitive on the (t + l)-arcs ofF The distinctive feature of the cubic case is that Aut(F) acts regularlyon the t-arcs

Proposition 181 Let [a] be a t-arc in a cubic t-transitive graph TThen an automorphism of F which fixes [a] must be the identity

Proof Suppose is an automorphism fixing each vertex laquo0 laquoigt bull bull bull gt regt-If is not the identity then does not fix all t-arcs in F It followsfrom Lemma 174 that there is some t-arc [] such that fixes []but does not fix both successors of [] Clearly if 0t-iu^1u^2) arethe vertices adjacent to then must interchange u^ and u^2 Letw ^ i be a vertex adjacent to 3o- Since F is t-transitive there is anautomorphism h euro Aut(F) taking the t-arc (w fio f3t-i) to [] andwe may suppose the notation chosen so that h((3t) = u^ bull Then hand fh are automorphisms of F taking the (t + l)-arc [w0 to its twosuccessors and by Theorem 175 Aut(F) is transitive on (t + l)-arcsThis contradicts our hypothesis and so we must have = 1 bull

From now on we shall suppose that we are dealing with a cubic t-transitive graph F and that we have chosen an arbitrary t-arc [a] in F

Symmetric graphs of degree three 139

If the stabilizer sequence of this t-axc is

Aut(r) = G gt Ft gt Ft-i gt gt FQ

then Proposition 181 implies that |F0| = 1 Consequently we know theorders of all the groups in the stabilizer sequence

111 = 2 ( 0 lt t lt t - l )

|Ft| = 3x2-1

G = n x 3 x 2-1 (n = |VT|)The structure of these groups can be elucidated by investigating cer-

tain sets of generators for them These generators are derived from thesets Yi defined for the general case in Chapter 17 Let at-iv^v^ bethe vertices adjacent to at and let gr (r = 12) denote automorphismstaking [a] to (ai ai at v^) We shall use the following notation

9 = 9i ô = 9T192 xi=g~lxag (i = 1 2 )

The effect of these automorphisms on the basic i-arc [a] is indicated inFigure 13 We note that these automorphisms are unique as a conse-quence of Proposition 181

bull bull [a]

-bull bull bull bull bull bull bull92 [laquo1 N^ xo[a]

Figure 13 the effect of 5132 and xo on [a]

In this chapter (X) will denote the subgroup of Aut(F) generated bythe set X

Proposition 182 The stabilizer sequence of a cubic t-transitive graphwith t gt 2 has the following properties(1) Fi = (xoXiXi-i) fori = 12(2) ifG = (xoxixt) thenGG lt 2

Proof We shall use the notation and results of Propositions 177 and178 In the cubic case we have Fo = 1 and the set Yi consists of theelements g^gi and their inverses g^ g for 1 lt j lt i

(1) It follows from part (2) of Proposition 177 that Fi = (Fi) Now


and so Ft = (xoxi xraquo-i)(2) It follows from Proposition 178 that the group G = (Yi+i) that

is (xo x bullxt) is a subgroup of index 1 or 2 in G provided that thegirth of F is greater than three If the girth is three then it is easy tosee that the only possibility is t = 2F = K4 and we may verify theconclusion explicitly in that case

(3) If G = G then (xog) contains (xoxi xt) = G mdash G If|G G = 2 then T is bipartite and each element g of G movesvertices of F through an even distance in F But the element g = gmoves some vertices to adjacent vertices and so g pound G Thus adjoiningg to G must enlarge the group and since G is a maximal subgroup ofG (because it has index 2) we have (Gg) = (xog) = G bull

In the previous chapter we considered Petersens graph obtaining forthe 3-arc [a] = (12341523) the automorphisms gx = (13)(245)2 =(13524) Hence

x0 = (34) Xl = (12) x2 = (35) x3 = (14)

We know that this graph is not bipartite since it has cycles of length 5and so in this case G = (xoxiX2X3) = G laquo 55

Another simple example is the 2-transitive graph Q3 the (ordinary)cube graph depicted in Figure 14 Taking [a] = (123) we have theautomorphisms as listed

5 (i

8 7

4 sFigure 14 the cube graph Q3

51 = (1234)(5678) g2 = (123785)(46)

xo - (36)(45) X = (16)(47) x2 =

In this case the graph is bipartite and G = (aroxiX2) preserves thebipartition

VQ3 = 1357U2468It follows that G G = 2


The main result on i-transitive cubic graphs is that there are no finiteexamples with t gt 5 The proof of this very important result is due toTutte (1947a) with later improvements by Sims (1967) and Djokovic(1972) Following these authors we shall obtain the result as an alge-braic consequence of the presentation of the stabilizer sequence given inProposition 182 A rather more streamlined proof using geometricalarguments to replace some of the algebraic calculations has been givenby Weiss (1974)

We shall suppose that t gt 4 as this assumption helps to avoid vacuousstatements We observe that each generator Xi (i gt 0) is an involutionand that each element of Fi (1 lt i lt t mdash 1) has a unique expression inthe form

xpxa xT where 0ltpltaltltTlti mdash 1

where we allow the empty set of subscripts to represent the identityelement The uniqueness of the expression is a consequence of the factthat there are 2l such expressions and Fi = 2l for 1 lt i lt t mdash 1

The key idea is to determine which stabilizers are abelian and whichare non-abelian It is immediate that Fi and F2 are abelian since |Fi | =2 and (i^t = 4 Let A denote the largest natural number such that Fis abelian

Proposition 183 Ift gt 4 then 2 lt A lt t + 2)Proof We have already remarked that A gt 2 Suppose that F =(XQXX-I) is abelian so that its conjugate g~t+x~1Fgt~x+1 thatis (xt-+i- bull bull xt) is also abelian If

A - l gt pound - A + lthen both these groups contain ZA-I and together they generate Ghence ZA-I commutes with every element of G Now g2 e G (sinceg e G and G G lt 2) and so

ZA-I = 9~2xx~i92 = x+iwhence x0 = X2- This is false given t gt 4 since IF3I gt |F2| and so wemust have

A - l lt i - A + l that is Alt-(lt + 2)

as claimed D

Proposition 183 gives an upper bound for A in terms of t We shallfind a lower bound of the same kind by means of arguments involvingthe commutators [ab] mdash a~1b~1ab of the canonical generators a Notethat since these generators are involutions we have

[XiXj] = (XiXj)2


Lemma 184 The generators x satisfy the following conditions(1) [xuXj] = 1 if j - i| lt A but xiXj ^ 1 if j - i = A(2) The centre of Fj = (xo bull bull bull Xj-i) is the group XJ- bull bull bull ^ A - I ) (A lt

3 lt 2A)(3) The commutator subgroup of F i + 1 is a subgroup of (xi Xj_i)= g-lFi^glltiltt-2)

Proof (1) We may suppose without loss that j gt i then [xiXj] =g~t[xoXj-i]gt and so [xiXj] = 1 if and only if xo and Xj-i commuteThe result follows from the fact that F = (xo bull bull bull XA-I) is the largestabelian stabilizer

(2) If the non-identity element x of Fj is written in the form

xpxa xT (0 lt p lt a lt lt T lt j - 1)

then x does not commute with xp+ Further if p + A lt j then xp+belongs to Fj Similarly x does not commute with xT~ and if r mdash A gtmdash 1 then xT- belongs to Fj Thus if x is in the centre of Fj thenp gt J mdash A and r lt A - 1 so that x is in (XJ- x^-i) Conversely itfollows from (1) that every element of this group is in the centre of Fj

(3) Provided that 1 lt i lt t - 2 the groups Fj = (x0 Xj_i)and g~lFig = (xixraquo) are different and they are both of indextwo in Fi+i and consequently normal in Fi+i Thus their intersection(xi Xi-i) = g~1Fiîg is normal in Fi+i and the quotient groupFi+i Ig~lFi-ig) is abelian since it has order 4 Hence the commutatorsubgroup of Fi+i is contained in g~1Fi-ig bull

Since [xo x] belongs to the commutator subgroup of Fx-i it follows(from part (3) of Lemma 184 with i = A) that [xoiX^] belongs to thegroup (xi x_i) In other words there is a unique expression

[xox]=xlixv (1 lt fi lt v lt A - 1)

L e m m a 185 With the above notation we have

(1) fj + gtt-l (2) 2X-igtt-l

Proof (1) Suppose that p + A lt t mdash 2 Then (by part (3) of Lemma184) the element [XOXM+A] of the commutator subgroup of FM +A+I

is contained in (xi x^+x-i)- The centre of (x i X ^ + A - I ) is thegroup (xM x) and since this contains both x and [xo x] it followsthat [XOXM+A] commutes with x and with [XQJXA] Also x^ commutes


with x^+x since J lt A mdash 1 Hence we have the following calculation

1 [xoxx][xo OM

= [XoXx]

This implies that xM+ commutes with [xoxx] = x^-Xv But thisis false since X^+A does not commute with poundM but does commute withany other term in the expression for [xogtpound]- Thus our hypothesis waswrong and i + A gt t mdash 1

(2) If 2A - u lt t mdash 2 then using arguments parallel to those in (1) wemay prove that [x2-vXo] commutes with xx-u and with [xx-vX2-v]also Xx-v commutes with xo since v gt 1 A calculation like that in (1)then implies that XQ commutes with

[x-v X2-u] = Xp+X-v bull bull bull Xx

which is false Hence 2A mdash is gtt mdash 1 bull

Theorem 186 (Tutte 1947a) There is no finite t-transitive cubicgraph with t gt 5

Proof If t is at least four then Proposition 183 tells us that A ltt-- 2) However the results of Lemma 185 show that t mdash 1 mdash A lt i ltis lt 2A mdash t + 1 that is A gt t mdash 1) Now if t gt 4 there is an integer Asuch that

| ( laquo - l ) lt A lt | ( t + 2)

only when t = 457 It remains to exclude the possibility t mdash 7 whichis done by means of the following special argument

If F is a 7-transitive cubic graph then the inequalities for A andis imply that A = 4 i = is = 2 thus [pound004] = pound2- Also by part(3) of Lemma 184 [rox5] belongs to the group ( i j 12^314) If thestandard expression for [pound035] actually contains xplusmn then we can write[xogtX5] = hxplusmn where h e (xix2x3) so that h commutes with XQ andX4 Hence

x2 - XQX^ = (x0x4)2 = (xohxt)2 = (xo(xox5)

2)2

= (x5x0x5)2 = x5xlx5 = 1

Since this is absurd [xoX5] = (xox5)2 must belong to (zipound203)

Now the original definitions show that XiX2 and X3 fix the vertex03 of the 7-arc [a] and so pound0X5(03) = pound5poundo(a3) = pound5(03) That isxo fixes pound5(03) Further since x5 fixes a but not a2 we have a 7-arc [0] mdash (x5(a3)x5(a2)aia2a3a4a5a6) in T The three vertices


adjacent to ai are aoa2 and pound5(02) and since XQ fixes aoai and a2

it must fix x^(a2) also Consequently XQ fixes the whole 7-arc [0] andthis contradicts Proposition 181 Hence t = 7 cannot occur bull

Goldschmidt (1980) proved an important extension of this result

The simplest example of a 5-transitive cubic graph is constructed asfollows Let the symmetric group SQ act on the 6 symbols a b c d e and take the vertices of a graph Q to be the 15 permutations of shape(ab) and the 15 permutations of shape (ab)(cd)(ef) Join two verticesby an edge if and only if the corresponding permutations have differentshape and they commute For instance (ab) is joined to the vertices(ab)(cd)(ef) (ab)(ce)(df) and (ab)(cf)(de) while (ab)(cd)(ef) is joinedto (ab) (cd) and (e) Clearly any automorphism of the group S$ is anautomorphism of fi and so

|Aut(fi)| = |AutS6| = 1440 = 30 x 3 x 24

as we expect for a 5-transitive cubic graph with 30 vertices We canverify that O is indeed 5-transitive by working out generators in termsof the following 5-arc

(06) (ab)(cd)(ef) (cd) (ae)(bf)(cd) (ae) (ae)(bd)(cf)If 7T is an element of 56 denote the corresponding inner automorphism(conjugation) of 56 by |TT| Then the generators for the stabilizer se-quence may be chosen as follows

xQ = (cd) i = (ab)(cd)(ef) x2 = |(aamp)|

x3 = |(o6)(c)(de)| x4 = |(c)|

The groups which occur in the stabilizer sequence are

F5 = 54xZ2 F4 = D8x Z2 F3 = (Z2)3

F2 = (Z2)2 F=Z2

Finally we may choose x5 so that G = (x0 x5) is isomorphic to5e and so G G = 2 in accordance with the fact that the graph isbipartite

Additional Results

18a A non-bipartite 5-transitive cubic graph A 5-transitive cubic graphwith 234 vertices which is not bipartite can be constructed as followsThe vertices correspond to the 234 triangles in PG(23) and two verticesare adjacent whenever the corresponding triangles have one commonpoint and their remaining four points are distinct and collinear Theautomorphism group is the group Aut PSL(33) of order 11232 = 234 x3 x 2 4


18b The sextet graphs (Biggs and Hoare 1983) Let q be an odd primepower Define a duet to be an unordered pair of points ab on the pro-jective line PG(lq) = GF(q) U oo and a quartet to be an unorderedpair of duets ab | cd such that the cross-ratio

^(a - d)(b - c)

(The usual conventions about oo apply here) A sextet is an unorderedtriple of duets ab | cd | ef such that each of ab | cd cd ef andef | ab is a quartet There are q(q2 mdash l)24 sextets if q = 1 (mod 4)and none if q = 3 (mod 4)

When q = 1 (mod 8) it is possible to define adjacency of sextetsin such a way that each sextet is adjacent to three others Thus weobtain a regular graph T(q) of degree 3 whose components poundo(lt) are allisomorphic The sextet graph S(p) is defined to be poundo(p) if p = 1 (mod8) and T0(p

2) if p = 357 (mod 8)The sextet graphs S(p) so defined form an infinite family of cubic

graphs one for each odd prime p The graph S(p) is 5-transitive whenp = 3 or 5 (mod 8) and 4-transitive otherwise The order of S(p)depends on the congruence class of p modulo 16 as follows

n = mdashpip2 mdash 1) when p = 115 (mod 16)

n = ^rp(p2 - 1) when p = 79 (mod 16)

n = ^p2ipA - 1) when p = 351113 (mod 16)

The group AutS(p) is PSL(2p) PGL(2p) PTL(2p2) in the respec-tive cases The two smallest 5-transitive sextet graphs are 5(3) which isisomorphic to the graph 0 described above and 5(5) which is a graphwith 650 vertices

18c Conway s presentations and the seven types Given an arbitrary t-arc [a] let a and b denote the automorphisms taking [a] to its successors(so a = pi and b = g2 in the notation described at the beginning of thischapter) Also let a be the automorphism which reverses [a] that is

o-(cti) = at-i (0ltilt t)

Since we know that Aut(f) acts regularly on the i-arcs it follows thata2 is the identity and aacr is either a1 or 61 We denote the case whencracr = a1 by t+ and the case when aaa = b~1 by t~ It turns out thatthe t+ case can occur only when t = 2345 and the t~ case only whent = l24

In each of the cases it can be shown by analysis of the action of


suitable combinations of ab and a on [a] that certain relations must

hold in Aut(F) For example in the 2+ case these relations are

a2 = 1 (era)2 = 1 (ab)2 = 1 (a^b)2 = 1 abaa2 = b2

In the 5 + case they are

a2 = 1 aaf = 1 (crb)2 = 1 (a1))2 = 1 ( a~V) 2 = 1

(a-363)2 = 1 a4b4a4 = ba a4baab = ba3b

Let us denote the groups generated by a b and ltr subject to theappropriate relations by

G2 G3 G4 Gh Gx G2 G4 bull

Each of these groups is an infinite group of automorphisms of the infinitecubic tree T3 acting regularly on the i-arcs for the relevant value of tand they are the only such groups up to conjugacy in Aut(Ta) Moredetailed information about the seven groups using different presenta-tions is given by Djokovic and Miller (1980) and Conder and Lorimer(1989)

18d Finite cubic graphs and groups Any group acting regularly on thei-arcs of a finite cubic graph F is a quotient of one of the seven groups in18c The quotient is defined by adding relations which represent cyclesin T a cycle of length I in T corresponding to a word of length I in a andb which represents the identity For example adding the relation a4 mdash 1to the relations for G j defines a group Gîa4) This is the group of thecube Qs as can be verified by showing that the permutations

o = (1234)(5678) b = (123785)(46) a = (13)(57)satisfy the defining relations for Gîa4) and represent automorphismsof Qz acting in the prescribed way on the 2-arc (123) (see Figure 14)

18e Coset enumeration In the notation of the Conway presentationsthe stabilizer of a f-arc is

Ft = (a-ibii= 12t)

If G is a quotient of G^ or GJ then the index G Ft is the cardinalityof a cubic graph for which G is a t-transitive group of automorphismsThe index may be finite or infinite but if it is finite the method of cosetenumeration will (in principle) determine its value This is a power-ful method for constructing finite ^-transitive cubic graphs See Biggs(1984a) for further details

18f The structure of a stabilizer sequence The groups occuring in the


stabilizer sequence are determined up to isomorphism as in the followingtable (Note that when pound = 24 both the t+ and t~ cases can occur butthe abstract groups are the same)

t Fl F2 F3 F4 F5

1 Z 3

2 Z 2 S3

3 Z 2 (Z2)2 D12

4 Z2 (Z2)2 As2 (Z2)2

)5 Z2 (Z2)2

18g Symmetric Y and H graphs Let Y and H denote the trees whosepictorial representations correspond to the respective letters Both ofthese trees have vertices of degree 1 (leaves) and 3 only Given any suchtree T we can form an expansion of T by taking a number n of disjointcopies of T and joining each set of corresponding leaves by a cycle oflength n each cycle has a constant step and different cycles will ingeneral have different steps For example when T = K2 we get thegraphs P(n t) described in 15e by joining one set of leaves with step 1and the other set with step t

Clearly an expansion of T is a cubic graph The result quoted in 15eimplies that only seven expansions of K2 are symmetric Horton andBouwer (1991) showed that there are only six other expansions whichare symmetric Four of them are expansions of Y n = 7 steps 124n = 14 steps 135 n = 28 steps 139 n = 56 steps 1925 Theother two are expansions of H n = 17 steps 1248 and n mdash 34 steps191315

18h Fosters census of symmetric cubic graphs (More details and bibli-ographical references relating to the following sketch are given by Bouwer(1988)) In 1920 two electrical engineers GA Campbell and RM Fos-ter wrote a paper in which the graph ^33 was used in the context oftelephone substation and repeater circuits Twelve years later Fosterpublished drawings of nine symmetric cubic graphs He continued towork on the subject and in 1966 he spoke at a conference at the Uni-versity of Waterloo where he distributed a mimeographed list of suchgraphs with up to 400 vertices In 1988 when Foster was just 92 Bouwerand his colleagues published Fosters census for graphs with up to 512vertices Remarkably only five graphs (out of 198) are known to havebeen missed by Foster and workers in this field are convinced that therecan be very few others if any

The graphs with n lt 30 vertices are as follows


Ki 33 Q3 Petersens graph Heawoods graph 5(7) P(83)(see 15e) the Pappus graph (see 19h) P(103) or the Desarguesgraph (see 19b) the dodecahedron P(125) Y(7 124) (see18g) and fi = 5(3)

18i All 5-transitive cubic graphs with less than 5000 vertices Cosetenumerations based on the Conway presentations and other techniqueshave established that the following list of 5-transitive cubic graphs withn lt 5000 vertices is almost certainly complete

n - 30 the sextet graph 5(3) group Gpound(a8)n = 90 a threefold cover of 5(3) (see 19c) group Gpound (610)n mdash 234 the graph described in 18a group Gg(a13)n mdash 468 a double covering of the previous graph group G$(b12)n = 650 the sextet graph 5(5) group Gpound(a12)n = 2352 a graph to be described in 19e group Gg(a14)n = 4704 a double covering of the previous graph group G$((ab)8)

18j The symmetric group 5io is a quotient of G$ (Conder 1987) Thefollowing permutations of 1 9 X satisfy the Conway relations forG j as given in 18c

a = (12) (34675) (89X) b = (1246853)(79X) a = (12)(34)(56)(9JsT)

Since these permutations generate the symmetric group 5io it followsthat there is a 5-transitive cubic graph with 10148 = 75600 verticesThe graph can be constructed in a way which shows that it is closelyrelated to the simplest 5-transitive cubic graph the graph fi = 5(3)(Lorimer 1989)

19

The covering graph construction

In this chapter we shall study a covering graph technique which incertain circumstances enables us to manufacture new symmetric graphsfrom a given one The method was first used in this context by JH Con-way who used the simple version discussed in Theorem 195 to show thatthere are infinitely many connected cubic graphs which are 5-transitiveThe general version given here was developed in the original 1974 edi-tion of this book and has since found several other applications some ofwhich are described in the Additional Results at the end of the chapterThe related technique of voltage-graphs (see Gross 1974) is much usedin the theory of graph embeddings

We shall use the symbol ST to denote the set of 1-arcs or sides of agraph F each edge u v of F gives rise to two sides (u v) and (vu)For any group K we define a K-chain on F to be a function ltjgt SF mdash Ksuch that ltj)(uv) = (^(iû))1 for all sides (uv) of F

Definition 191 The covering graph F = T(K ltp) of F with respectto a given Jif-chain 0 on F is defined as follows The vertex-set of F isK x VT and two vertices (KI wi) K2 V2) are joined by an edge if andonly if

(vitV) euro SF and K2 mdash Ki4gtv v2)-

It is easy to check that the definition of adjacency depends only on theunordered pair of vertices

As an example let F = K4 and let K be the group Z2 whose elements


r f

Figure 15 Q3 as a double covering of K4

are 1 and z the function ltgt which assigns z to each side of K4 is a Z2-chain on K4 The covering graph F(Z2 ltjgt) is isomorphic to the cube Q3as depicted in Figure 15

Suppose that a group G acts as a group of automorphisms of a groupK that is for each g in G we have an automorphism g of K such thatthe function g raquomdashgt p is a group homomorphism from G to Autif In thissituation we define the semi-direct product of K by G denoted by KxGto be the group whose elements are the ordered pairs (ng) with thegroup operation given by

Let T be a graph cjgt a if-chain on T and let G = Aut(F) Then Gacts on the sides of T by the rule g(uv) = (g(u)g(v)) and we maypostulate a special relationship between the action of G on K and itsaction on ST

Definition 192 The ftT-chain ltjgt is compatible with the given actionsof G on K and ST if the following diagram is commutative for each ginG

KST

ST -1+ K

Proposition 193 Suppose that T is a graph whose automorphismgroup G = Aut(F) acts as a group of automorphisms of a group KSuppose further that there is a K-chain ltfgt onT which is compatible withthe actions of G on K and ST Then the semi-direct product KxG is agroup of automorphisms of the covering graph T = T(K ltfgt)

The covering graph construction 151

Proof Define the effect of an element (laquo g) of Kx G on a vertex (laquo v)of F by the rule

(K9)(KV) ~ (Kg(n)g(v))

Using the definition of compatibility a simple calculation shows thatthis permutation of VT is an automorphism of F bull

The usefulness of the covering graph construction lies in the fact thata much stronger version of Proposition 193 is true

Proposition 194 With the notation and hypotheses of Proposition193 suppose also that G is transitive on the t-arcs ofT Then KxG istransitive on the t-arcs ofF

Proof ^ Let ((KOVQ) (Ktvt)) and ((KOVO) (laquoJgtut)) b e t w o l~arcs in F Then (vo bull vt) and (vo vt) are f-arcs in F and so thereis some g in G such that g(vi) = v (0 lt i lt t) Suppose we chooseK in K such that (ng) takes (KOVQ) to (KOVO) that is we chooseK = KQI^KQ))1 Then we claim that (ng) takes niVi) to (K^^)

fo ra lHeuro0 l 0 -The claim is true when i = 0 and we make the inductive hypothesis

that it is true when i mdash j - 1 so that(laquoj-_iuj_i) = (K5)(KJ_IVJ_I) = ^giKj^givj-i))

Since (KJVJ) is adjacent to (KJ-IVJ-I) we have Kj mdash KJ-I4gt(VJ-IVJ)

and the corresponding equation holds for the primed symbols as wellThus

Consequently (laquolt) takes (KJVJ) to (K^^-) and the result follows bythe principle of induction bull

The requirement that a compatible if-chain should exist is ratherrestrictive In fact for a given graph F and group K it is very likely thatthe only covering graph is the trivial one consisting of K componentseach isomorphic with F However it is possible to choose K (dependingon F) in such a way that a non-trivial covering graph always exists

Let us suppose that a t-transitive graph F is given We define K to bethe free Z2-module on the set poundT thus K is the direct product of ETcopies of 2 and its elements are the formal products Ylea raquo wherea(a) = 0 or 1 and the product is over all ea in ET The automorphismgroup G = Aut(F) acts on K through its action on ET and furthermore


there is a fC-chain (fgt onT defined by the rule 4gtuv) = e i where e =u v regarded as an element of K This K-chain is compatible withthe actions of G on K and ST and so the covering graph F = T(K ltjgt)exists and (by Proposition 194) its automorphism group is transitive oni-arcs bull

Theorem 195 Let T be a t-transitive graph whose rank and co-rankare r(T) and s(T) Then with the special choices ofK and ltfgt given abovethe covering graph T consists of2r^ connected components each having2a(r) |VT| vertices

Proof Pick a vertex v of F and let Fo denote the component of Fwhich contains the vertex (lv) If

V - U0Ui U[ = V

are the vertices of a cycle in F with edges e = UJ_I Ui then we havethe following path in Fo

(lv) (eiux) (eie2u2) ( e i e 2 -etv)

Conversely the vertex (n v) is in Fo only if laquo represents the edges of acycle in F Since there are s(F) independent cycles in F there are 2Sêlements K in K such that (K V) is in Fo- It follows that Fo has 2s^r- | VTvertices further F is vertex-transitive and so each component has thisnumber of vertices Finally since

|VT | = KVT = 2 lE r | |FF| and r(F) + s(F) = |JET|

there must be 2 r^ r components D

Corollary 196 There are infinitely many cubic 5-transitive graphs

Proof We know that there is at least one cubic 5-transitive graphthe graph ft constructed at the end of the previous chapter Applyingthe construction of Theorem 195 to O we obtain a cubic 5-transitivegraph fi0 with 2s^n^|FJ7| vertices and since s(Cl) gt 0 this graph is notisomorphic with il We may repeat this process as often as we pleaseobtaining an infinite sequence of graphs with the required properties

bull

Of course the number of vertices used in Corollary 196 quickly be-comes astronomical for instance the two graphs which follow fi in thesequence have about 221 and 2100000 vertices respectively Biggs andHoare (1983) have given an explicit construction for infinitely many cu-bic 5-transitive graphs which involves much smaller numbers (see 18b)


Additional Results

19a Double coverings Let G be the automorphism group of a con-nected graph T and let G act on the group Z2 by the rule that g is theidentity automorphism of Z2 for each g in G Then the Z2-chain ltfi on Fwhich assigns the non-identity element of Z2 to each side of F is compat-ible with the actions of G on SV and Z2 The covering graph F(Z2 (fgt)is connected if and only if F is not bipartite For example applying theconstruction to the graph with 234 vertices described in 18a we get aconnected 5-transitive cubic graph with 468 vertices

19b The Desargues graph The construction of 19a applied to Pe-tersens graph results in a cubic 3-transitive graph with 20 vertices Thevertices of this graph correspond to the points and lines in the Desarguesconfiguration with two vertices being adjacent if they correspond to anincident (point line) pair This graph was described by Coxeter (1950)together with several others derived from geometrical configurations

19c A threefold covering of 5(3) The second 5-transitive cubic graphin order of magnitude is a graph with 90 vertices which is a threefoldcovering of the sextet graph 5(3) (see 18i) Ito (1982) constructed anexplicit Z3-chain on 5(3) which shows that the graph is a covering graphof 5(3) in the sense of this chapter

19d Another covering construction for 5-transitive cubic graphs Sup-pose that F is a cubic graph and G = Aut(F) is a group of type 4+ Then the automorphism a~lb fixes the vertices ao ai ai and 03 of thebasic 4-arc [a] and (because the degree is 3) it must fix the other ver-tices 3i32 adjacent to oca2 respectively By considerations of orderwe see that this is the only non-identity automorphism with this prop-erty So for each e = v w euro ET the group Lvw has order 2 in otherwords there is a unique involution je which fixes e and the four verticesadjacent to e The involutions je generate the group G (Proposition182) which is normal of index 1 or 2 in G Consequently G acts byconjugation as a group of automorphisms of G

If we take K = G and define a If-chain on F by

ltfgt(vw)=je

then the compatibility condition is satisfied and by Proposition 194we have a graph F on which KxG acts 4-transitively However there isa bonus As shown by Biggs (1982b) there is an extra automorphismF so that F is in fact a 5-transitive graph


19e A 5-transitive cubic graph with 2352 vertices The simplest caseof 19d is when F = 5(7) a graph with 14 vertices also known as theHeawood graph In this case both F and its 5-transitive covering graph Fwith 2352 vertices can be constructed directly in terms of the seven-pointplane PG(22) (Biggs 1982a)

19f Conway generators for the covering graph Let a be the Conwaygenerator for the t-transitive group G of F with respect to the i-arc[a] and suppose ltgt is a compatible if-chain Then the correspondinggenerator a for the group KxG of T(K ltjgt) is (A a) where A = ltgt(co oi)

19g Homological coverings Let F be a graph with co-rank s and let Rbe a ring The first homology group with coefficients in K1 of a graph F isthe direct product R of s copies of R (This a just a mild generalizationof the cycle space denned in Chapter 4) The functorial properties ofhomology imply that the automorphism group of F acts as a group ofautomorphisms of the homology group and so a covering graph F can beconstructed using K = Rs Biggs (1984b) gave an explicit form of thisconstruction and showed that when R = TL the number of componentsof F is equal to the tree-number of F

19h The Pappus graph In the homological covering construction takeR mdash Z 3 as the coefficient group and F = 33 so that s = 4 andK = Z3 The covering graph in this case has 27 components eachwith 6 x 3427 = 18 vertices Each component is a copy of the Pappusgraph whose vertices correspond to the points and lines of the Pappusconfiguration with adjacent vertices corresponding to an incident (pointline) pair See also Coxeter (1950)

20

Distance-transitive graphs

In Chapter 15 a connected graph F was defined to be distance-transitiveif for any vertices uvxy of F satisfying d(u v) = d(xy) there is anautomorphism g of F which takes a t o i and v to y

ro(raquo) r(raquo) rs(tgt)

Figure 16 a distance-partition of K^

It is helpful to recast the definition For any vertex v of a connectedgraph F we define

where i is a non-negative integer not exceeding d the diameter of F Itis clear that F0(igt) = v and VT is partitioned into the disjoint subsetsro(v) Td(v) for each v in VT Small graphs may be depicted in amanner which emphasises this partition by arranging their vertices incolumns according to distance from an arbitrary vertex v For example^33 is displayed in this way in Figure 16


Lemma 201 A connected graph F with diameter d and automor-phism group G = Aut(F) is distance-transitive if and only if it is vertex-transitive and the vertex-stabilizer Gv is transitive on the set Fj(igt) foreach i euro 0 1 d and each v euro VT

Proof Suppose that F is distance-transitive Taking u = v and x = yin the definition (as given above) we see that F is vertex-transitiveTaking y = v we see that Gv is transitive on Ti(v) (0 lt i lt d)

Conversely suppose vertices u v x y are given such that d(u v) =d(x y) mdash i Let g be an automorphism such that g(v) = y and leth 6 Gy be such that h(g(u)) = x Then hg takes u to x and v to y

D

As we shall see the adjacency algebra (defined in Chapter 2) plays amajor part in the study of distance-transitive graphs In preparation forthe algebraic theory we begin by investigating some simple combinatorialconsequences of the definition

For any connected graph F any vertices uv of F and any non-negative integers h and i define Shi(u v) to be the number of verticesof F whose distance from u is h and whose distance from v is i That is

8hiuv) = w e VT | d(uw) = h and d(vw) = i

In a distance-transitive graph the numbers Shiu v) depend not on theindividual pair (u v) but only on the distance d(u v) So if d(u v) = jwe shall write

Shij = Shi(uv)

Definition 202 The intersection numbers of a distance-transitivegraph with diameter d are the numbers Shij where hi and j belong tothe set 01 d

Clearly there are (d + I)3 intersection numbers but it turns out thatthere are many identities relating them and in due course we shall showthat just 2d of them are sufficient to determine the rest

Consider the intersection numbers with h = 1 For a fixed j SUJ isthe number of vertices w such that w is adjacent to u and d(v w) = iwhen d(u v) mdash j Now if w is adjacent to u and d(u v) = j thend(v w) must be one of the numbers j mdash ljj + 1 in other words

a i i j = 0 if i^j-ljj + l

More generally s^j = 0 if the largest of hij is greater than the sumof the other two

Distance-transitive graphs 157

For the intersection numbers SUJ which are not identically zero weshall use the notation

CJ = 8lj-lj a3 mdash sljjgt fy = Sij+itj

where 0 lt j lt d and it is convenient to leave c$ and bd undefinedThe numbers Cjaj bj have the following simple interpretation in termsof the diagrammatic representation of F introduced at the beginning ofthis chapter If we pick an arbitrary vertex v and a vertex u in Tj(v)then u is adjacent to Cj vertices in Tj-i(v)aj vertices in Tj(v) and bjvertices in Fj+i (y) These numbers are independent of u and v providedthat d(u v) = j

Definition 203 The intersection array of a distance-transitive graphis

C i Cj

a0 ai a bo bi bj

For example consider the cube Qz which is a distance-transitivegraph with diameter 3 From the representation in Figure 17 we maywrite down its intersection array

1 2 3t(Q3) = 0 0 0 0

3 2 1

Figure 17 Qz as a distance-transitive graph

We observe that a distance-transitive graph is vertex-transitive andconsequently regular of degree k say Clearly we have bo = k andao = 0 C mdash 1 Further since each column of the intersection arraysums to k if we are given the first and third rows we can calculatethe middle row Thus it is both logically sufficient and typographicallyconvenient to use the alternative notation

t(r) = fc amp ampd_ilC2Cd

However the original notation of Definition 203 is intuitively helpful


and we shall continue to use it whenever it seems appropriate In duecourse we shall see that the intersection array determines all the inter-section numbers Shij

Many well-known families of graphs are distance-transitive althoughthis apparent profusion of examples is rather misleading because theproperty is in some senses very rare The complete graphs Kn and thecomplete bipartite graphs Kkk are distance-transitive Their diametersare 1 and 2 respectively and the intersection arrays are

f 1 ) ( I ktKn) ~ 0 n - 2 i(Kkik) = 0 0 0

[n-1 J [k fc-1 The triangle graphs At = L(Kt) (p 21) are distance-transitive withdiameter 2 and for t gt 4

1 4

0 t-2 2 - 82f - 4 t - 3

Many other distance-transitive graphs will be described in the followingchapters

Denote by ki (0 lt i lt d) the number of vertices in Fj(u) for anyvertex v in particular ampo = 1 and k = kProposition 204 Let T be a distance-transitive graph whose intersec-tion array is kb bd-i 1C2 Cd Then we have the followingequations and inequalities(1) ki^lH-i = ha (1 lt i lt d)(2) 1 lt c2 lt c3 lt lt cd(3) kgth gtb2 gtgtbd-iProof (1) For any v in VT there are fcj_i vertices in Fi-i(v) and eachis joined to 6j_i vertices in Fi(v) Also there are ampj vertices in Ti(v) andeach is joined to Cj vertices in Fj_i(i) Thus the number of edges withone end in Fi_i(t) and one end in Fi(v) is fc_iampt_i = kiCi

(2) Suppose u is in Ti+i(v) (1 lt i lt d - 1) Pick a path vxu oflength i + l then d(xu) mdashi liw is in Fi_i(a)nr1(u) then d(vw) = iand so w is in Fi(v) n Fi(u) It follows that

Ci = tri_i(a) n Fi(u)| lt |Fi(t) n Ti(u) = ci+1(3) This is proved by an argument analogous to that used in (2) bull

Proposition 204 provides some simple constraints which must be sat-isfied if an arbitrary array is to be the intersection array of some distance-transitive graph We shall obtain much more restrictive conditions in the


next chapter However in order to derive these conditions we need notpostulate that the graph is distance-transitive but merely that it hasthe combinatorial regularity implied by the existence of an intersectionarray This is the justification for the following definition

Definition 205 A distance-regular graph is a regular connected graphwith degree k and diameter d for which following holds There arenatural numbers

bo = k bibd-i c mdash 1 C 2 C d

such that for each pair (u v) of vertices satisfying d(u v) = j we have

(1) the number of vertices in rj_1(t)) adjacent to u is Cj (1 lt j lt d)(2) the number of vertices in Fj+i (v) adjacent to u is bj (0 lt j lt d mdash 1)

The array kbi ampltj_i 1C2 Cd is the intersection array of F

Note that a distance-regular graph with diameter d = 2 is simply astrongly regular graph as denned in 3c In terms of the general defini-tion the parameters a and c of a strongly regular graph are given bya mdash k mdash l mdash bi and c = cltx-

It is clear that a distance-transitive graph is distance-regular but theconverse is not true Although many familiar examples of distance-regular graphs are distance-transitive it is possible to construct arbi-trarily large families of distance-regular graphs which are not distance-transitive Several examples will be given in the course of the followingchapters

We shall now construct a basis for the adjacency algebra of a distance-regular graph Given a graph F with vertex-set v vn and diame-ter d define a set Ao A i Ad o fnxn distance matrices as follows

1 X dvrva) = h0 otherwise

In particular Ao = I and Ai is the usual adjacency matrix A of F Wenotice that Ao + Ai + + Ad = J where J is the all-1 matrix

-Imdash 1

Lemma 206 Let F be a distance-regular graph and let

kh bd-i 1 c 2 ca

be its intersection array For 1 lt i lt d mdash 1 define a = k mdash bi mdash Ci then

AAi = 6pound_iAi_i + aAi + Ci+iAi+i (1 lt i lt d - 1)

Proof Prom the definition of A and Ai it follows that (AAj)rs isthe number of vertices w of F such that d(vrw) = 1 and d(vsw) =i If there are any such vertices w then d(vr vs) must be one of thenumbers i mdash 1 i i +1 and the number of vertices w in these three cases


is 6i_iOiCi+i respectively Thus (AA)rs is equal to the (rs)-entryof the matrix on the right-hand side bull

Theorem 207 (Damerell 1973) Let F be a distance-regular graphwith diameter d Then Ao A x Ad is a basis for the adjacencyalgebra AT) and consequently the dimension of A(T) is d+1

Proof By recursive applications of the lemma we see that Aj is apolynomial Pi(A) for i = 2d The form of the recursion showsthat the degree of pi is at most i and since Ao A i A^ are linearlyindependent (exactly one of them has a non-zero entry in any givenposition) the degree of p is exactly i

Since Ao + Ai + 4- Ad = 3 and T is fc-regular we have

(A - AI)(Ao + Ai + + Ad) = 0

The left-hand side is a polynomial in A of degree d+1 so the dimensionof -4(r) is at most d+1 However since Ao A j A^ is a set of d+1linearly independent members of A(T) it is a basis and the dimensionis equal to d + 1 bull

It follows from Theorem 207 that a distance-regular graph has justd+1 distinct eigenvalues the minimum number possible for a graph ofdiameter d These eigenvalues and a remarkable formula for calculatingtheir multiplicities form the subject of the next chapter

The full set of (d + I)3 intersection numbers can be defined for adistance-regular graph this is a trivial remark for a distance-transitivegraph but it requires proof in the distance-regular case In the courseof the proof we shall relate these intersection numbers to the basis

Proposition 208 Let F be a distance-regular graph with diameter d(1) The numbers Sih(uv) hi euro 01 d depend only on d(uv)(2) If Shi(uv) = Shij when d(uv) = j then

d

j=o

Proof We prove both parts in one argument Since Ao A 1 is a basis for A(T) the product AÂj is a linear combinationNow

rs mdash Shi(vrvs)

and there is just one member of the basis whose (r s)-entry is 1 itis that Aj for which d(vrvs) = j Thus Shi(vrvs) = thij and so


Shivrvs) depends only on d(vTvs) Further the coefficient thij is justthe intersection number Shij- Q

At this point a few historical remarks are in order The theory whichunderlies our treatment of the adjacency algebra of a distance-regulargraph was developed in two quite different contexts First the associ-ation schemes used by Bose in the statistical design of experiments ledto an association algebra (Bose and Mesner 1959) which correspondsto our adjacency algebra Bose and others also studied strongly regulargraphs which as we have noted are just distance-regular graphs withdiameter 2 Secondly the work of Schur (1933) and Wielandt (1964) onthe commuting algebra or centralizer ring of a permutation group cul-minated in the paper of Higman (1967) which employs graph-theoreticideas very closely related to those of this chapter The discovery ofsporadic simple groups as the automorphism groups of strongly regu-lar graphs (for example by Higman and Sims (1968)) gave a powerfulimpetus to work in this area The formulation in terms of the proper-ties of distance-transitivity and distance-regularity was developed by thepresent author and some of his colleagues in the years 1969-1973 anda consolidated account appeared in the first edition of this book (1974)In the last twenty years an extensive literature has been accumulatingThe reader is referred to the now-standard text of Brouwer Cohen andNeumaier [BCN] which admirably covers the state of the art up to 1989and contains a bibliography of 800 items

Additional Results

20a The cube graphs The fc-cube Qk is the graph defined as followsthe vertices of Qk are the 2fc symbols (eiC2 bull bull bull Cfc) where e = 0 or1 (1 lt i lt A) and two vertices are adjacent when the symbols differin exactly one coordinate The graph Qk (k gt 2) is distance-transitivewith degree k and diameter k and the intersection array is

~ K ^ tv trade J j K trade Z j bull bull j i 1 ] Z ) O ) laquo bull bull J C J

20b The odd graphs yet again The odd graphs Ok (k gt 2) are distance-transitive with degree k and diameter k mdash 1 The intersection array inthe cases k = 21 mdash 1 and k = 21 respectively is

21 - 121 - 221 - 2 1 + 11 + 11 1122 1 - 11 - 1

2121 - 121 - 1 I + 11 + 1 1122 I - 11 - 1


20c A distance-regular graph which is not distance-transitive Let ^denote the graph whose vertices are the 26 symbols a 6 (where i is aninteger modulo 13) and in which

di and aj are adjacent bullampbull i mdash j = 134

bi and bj are adjacent laquobull i mdash j mdash 256

a and bj are adjacent lt=gt i mdash j = 0139Then $ is distance-regular with diameter 2 and its intersection arrayis 106 14 But ty is not distance-transitive in fact there is no au-tomorphism taking a vertex a to a vertex bj (Adelson-Velskii et al1969)

20d Strengthening the distance-transitivity condition A connected sim-ple graph is r-ply transitive if for any two ordered r-tuples of vertices(xixr) and (yiyr) satisfying dxuxj) - (ampbull) for all ijthere is an automorphism g for which g(xi) = yi (1 lti lt r) Clearly a1-ply transitive graph is vertex-transitive and a 2-ply transitive graphis distance-transitive Meredith (1976) showed that the only 3-ply tran-sitive graphs with girth greater than 4 (equivalently ci = 1) are thecycles

20e 6-ply transitive graphs (Cameron 1980) The following is a com-plete list of all 6-ply transitive graphs(i) The complete multipartite graphs with parts of equal size (includingthe complete graphs as the case when the parts have size 1)(ii) The complete bipartite graphs with the edges of a complete matchingdeleted(iii) The cycles(iv) L(K3t3)(v) The icosahedron(vi) The graph whose vertices are the 3-subsets of a 6-set two verticesbeing adjacent whenever they have two common members

20f Strongly regular graphs and partial geometries A partial geometrypg(s t a) is an incidence structure of points and lines such that everyline has s +1 points every point is on t +1 lines two distinct lines meetin at most one point and for every non-incident (point line) pair (p I)there are a lines through p that meet I The graph whose vertices are thepoints two being adjacent if they are collinear is strongly regular withparameters k = s(t +1) a mdash t(a mdash 1) + s mdash 1 c = a(t +1) Equivalentlyit is a distance-regular graph with intersection array

s(t + 1) (s - a + l)t 1 a(t + 1)


20g Symmetric designs as distance-regular graphs A symmetric designwith parameters (v fc A) is a set P of points and a set B of blocks suchthat P mdash B = v each block has k points and each point is in fc blocksand each pair of points is in A blocks It follows from the definition that(v mdash 1)A = k(kmdash 1) When A = 1 a symmetric design is called a protectiveplane

The graph whose vertices are the points and blocks of a symmetricdesign two being adjacent when they are incident is distance-regularwith intersection array

^CA I K A 1 A fe j

For example when A = 1 we have the incidence graph of a projectiveplane the case fc = 3 is Heawoods graph 5(7) mentioned in 18h Ifthe projective plane is Desarguesian (that is if it can be coordinatizedusing a finite field) then the corresponding graph is distance-transitive

20h The classification problem for DT and DR graphs For each k gt 3there are only finitely many DT graphs with degree k This has beenproved in several ways see Cameron (1982) and Weiss (1985) for ex-ample For DR graphs the result has been established only in the casefc = 3 (Biggs Boshier and Shawe-Taylor (1986) see 21i)

For the general DR case the problem is to find an upper bound forthe diameter d in terms of fc Such a result could be regarded as astrengthening of the monotonicity conditions (2) and (3) of Proposition204 in which we seek to bound the number of repeated values amongthe columns (c aibi) of the intersection array An important result onthese lines was obtained by Ivanov (1983)

21

Feasibility of intersection arrays

In this chapter we shall study the following question Suppose that anarbitrary array of integers kbi bd-i 1 C2 c^ is given whenis there a distance-regular graph with this as its intersection array

The results obtained in the previous chapter provide some simple nec-essary conditions For example part (1) of Proposition 204 yields anexplicit formula for the numbers hi = Ti(v)

h == (kh k-Otecs a) (2 lt i lt d)These numbers must be integers so we have a non-trivial constraint onthe intersection array Similarly the monotonicity conditions in parts(2) and (3) of Proposition 204 must be satisfied

There are also some elementary parity conditions Let n = 1 + ampi + + kd be the number of vertices of the putative graph then if k is oddn must be even That is nk = 0 (mod 2) Similarly considering theinduced subgraph defined by the vertices in I(v) we see that feoj 5 0(mod 2) for 1 lt i lt d where Oj = k mdash 6j mdash Cj

These conditions are quite restrictive yet they are satisfied by manyarrays which are not realised by any graph For example 321 113passes all these tests and would represent a graph with degree 3 diam-eter 3 and 12 vertices In this case simple (but special) arguments canbe used to prove that there is no graph The main result of this chapteris a general condition which rules out a multitude of examples of thiskind

Recall that the adjacency algebra A(T) of a distance-regular graph T

Feasibility of intersection arrays 165

has as a basis the d+1 distance matrices A0) A x Ad which satisfyAftAj = J2 ShijAj This equation can be interpreted as saying that left-multiplication by Ah regarded as a linear mapping of A(T) with respectto the given basis is faithfully represented by the (d+ l)x(d+1) matrixBfc defined by

(This representation seems natural for our purposes although it is thetranspose of the one most commonly employed Since the algebra A(F)is commutative the difference is immaterial) The existence of thisrepresentation is sufficiently important to justify a formal statement

Proposition 211 The adjacency algebra A(T) of a distance-regulargraph F with diameter d can be faithfully represented by an algebra ofmatrices with d+1 rows and columns A basis for this representation isthe set BoBi Bd whereforhij euro 0 ld

)ij is the intersection number

bullThe members of -4(F) can now be regarded as square matrices of size

d+1 (instead of n) a considerable simplification What is more thematrix Bi alone is sufficient To see this we notice first that since(Bi)jj = suj the matrix Bi is tridiagonal

TO 1k d i C2

h a2 bull

bull bull cd

bull adA

We shall often write B for Bi and refer to B as the intersection matrixof F Note that it is just another way of writing the intersection arrayNow since the matrices Bj are images of the matrices Ai under a faithfulrepresentation the equation obtained in Lemma 206 carries over

BBi = 6i-iBi_i + aiBi + c i + iB i + 1 (1 lt i lt d - 1)Consequently each B is a polynomial in B with coefficients which de-pend only on the entries of B It follows from this (in theory) that A(T)and the spectrum of F are determined by B which in turn is determinedby the intersection array t(F) We shall now give an explicit demonstra-tion of this fact

Proposition 212 Let T be a distance-regular graph with degree k anddiameter d Then V has d+1 distinct eigenvalues k mdash Q AI A which are the eigenvalues of the intersection matrix B


Proof We noted in Chapter 20 that T has exactly d + 1 distincteigenvalues Since B is the image of the adjacency matrix A under afaithful representation the minimum polynomials of A and B coincideand so the eigenvalues of A are the same as those of B D

Each eigenvalue A common to A and B is a simple eigenvalue of Bsince B is a matrix of size d + 1 However the multiplicity m(A) of Aas an eigenvalue of A will usually be greater than one since the sum ofthe multiplicities is n the number of vertices We shall show how m(A)can be calculated from B alone

Let us regard A as an indeterminate and define a sequence of polyno-mials in A with rational coefficients by the recursion

vo() = 1

d+lvi+i) + (OJ - A)vi(A) + bi-iVi-iX) = 0 (i = 12d ~ 1)

The polynomial Vi() has degree i in A and comparing the definitionwith Lemma 206 we see that

Another interpretation of the sequence UJ(A) is as follows If weintroduce the column vector v(A) = [laquoo(A)ui(A) ^(A)] then thedefining equations are those which arise when we put to(A) = 1 and solvethe system Bv(A) = Av(A) using one row of B at a time and stoppingat row d mdash 1 The last row of B gives rise to an equation representingthe condition that v(A) is an eigenvector of B corresponding to theeigenvalue of A The roots of this equation in A are the eigenvaluesAo Ai Ad of B and so a right eigenvector v corresponding to Ajhas components (VJ)- = Vj(Xi)

It is convenient to consider also the left eigenvector Uj correspondingto AJ this is a row vector satisfying UjB = AjUj We shall say that avector x is standard when XQ mdash 1

Lemma 213 Suppose that m and Vj are standard left and right eigen-vectors corresponding to the eigenvalue Aj o B Then (VJ)J = kj(ii)jfor alii je 01 d

Proof Each eigenvalue of B is simple and so there is a one-dimensionalspace of corresponding eigenvectors It follows that there are uniquestandard eigenvectors u and v (If (u)o or (vi)0 were zero then thetridiagonal form of B would imply that uraquo = 0 Vj = 0)

Let K denote the diagonal matrix with diagonal entries fco fci bull feÛsing the equations bi-ikî = cfcj (2 lt i lt d) we may check that BK


is a symmetric matrix that is

BK = (BK)( - KB

Thus if UjB = XiUi (0 lt i lt d) we have

BKu| = KBul = K(uiB)t = K(Ain) = AltKuJIn other words Ku- is a right eigenvector of B corresponding to A Also(Ku)o = 1 and so by the uniqueness of v$ it follows that Ku = v

D

We notice that when i ^ I the inner product (uv) is zero since

Ai(Uj Vj) = UjBvj = Aj(Uj Vj)

Our main result is that the inner product with i = I determines themultiplicity m(Aj)

Theorem 214 With the notation above the multiplicity of the eigen-value Aj of a distance-regular graph with n vertices is

^) ioltiltd)

Proof For i = 0 1 d defined

3=0

We can calculate the trace of Lj in two ways First the trace of Aj iszero (j 7 0) and Ao = I so that

tr(L4) = (ui)otr(I) = n

On the other hand since Aj = u-(A) the eigenvalues of Aj are Vj(X0) Vj(Xd) with multiplicities m(A0) m(Xlti) consequently the traceof Aj is poundm(Aj)uj(A0- Thus

which gives the required result D

In the context of our question about the realisability of a given ar-ray we shall view Theorem 214 in the following way The numbersn(ui Vj) which are completely determined by the array represent mul-tiplicities of the eigenvalues of the adjacency matrix of a supposed graphand consequently if there is such a graph they must be positive integersThis turns out to be a very powerful condition


Definition 215 The array k bit bd-i 1 c2 Cd is feasible ifthe following conditions are satisfied

(1) The numbers kt = (kbi ampi-i)(c2c3 Cj) are integers (2 lt i lt d)(2) fc gt 6i gt gt 6d_i and 1 lt c2 lt lt cd(3) If n - 1 + k + amp2 + bull bull + kd and ltn = k - bt - a (1 lt i lt d - 1)ad mdash k mdash Cd then nk = 0 (mod 2) and fcaj = 0 (mod 2)(4) The numbers n(u$ vraquo) are positive integers (0 lt i lt d)

It should be noted that the definition of feasibility given above isa matter of convention The conditions stated are not sufficient forthe existence of a graph with the given array and indeed there aremany other independent feasibility conditions Some useful ones aregiven in 21c 21d and 21e the standard reference [BCN] providesa comprehensive treatment The four conditions which comprise ourdefinition of feasibility are chosen because they are particularly usefuland any reasonable way of testing a given array will surely include them

The four conditions are easy to apply in practice The calculation ofn(uj Vi) is facilitated by Lemma 213 which implies that

n3

For example consider the array 321 113 which as we have al-ready noted satisfies the first three conditions The eigenvalues of Bare 3 -1 and the roots of the quadratic equation A2 + A mdash 3 = 0 If0 is one of the quadratic eigenvalues the corresponding eigenvector is[10-0-1] and the multiplicity is

12 ( l + J + J + I) = 24(3 + deg2) = 24(6 0)which is clearly not an integer Thus there is no graph with the givenarray

For a positive example consider the array 2rr mdash 1 14 (r gt 2)for which the corresponding B matrix is

0 1 02r r 40 r - 1 2r - 4

It is easy to verify that k = 2r k2 = rr - 1) n = r + l)(r + 2) sothat conditions (1) (2) and (3) of Definition 215 are fulfilled


The eigenvalues of B are Ao = 2r Ai = r - 2 A2 = - 2 and thecalculation of the multiplicities goes as follows

v0 =

12r

gt(r-l) Vi =

1r - 21 - r

v2 = 1 - 21

m(Ai) n(ui

m(A2) =

1 + (r - 2)22r + (1 - r)2rr - 1)

1 i r ( r _ 1 N = 2~(r ~ 1)(r(u2v2) 1-

Since these values are integers condition (4) is satisfied and the array isfeasible In fact the array is realized by the triangle graph Ar+2 as wenoted in Chapter 20 (The eigenvalues and multiplicities of this graphwere found in a different way in Chapter 3)

Another example is the graph S representing the 27 lines on a cubicsurface (Chapter 8 p 57) This is a distance-regular graph with diam-eter 2 and intersection array 165 18 from which we may calculatethe spectrum

_ _ 16 4 -2s

Spec pound = ^ x 6 2 0

These examples have diameter 2 and so they are strongly regulargraphs In that case the multiplicities can also be obtained by moreelementary methods (see 3d) But for a general distance-regular graphthe multiplicity formula is invaluable

Additional Results

21a The spectra oQk and the Hamming graphs The eigenvalues of thefc-cube Qk are Araquo = k - 2i (0 lt i lt k) with multiplicities m(Xi) - ()

The fc-cube is the case q = 2 of the Hamming graph H(dq) whosevertices are the qd d-vectors with elements in a set of size q two beingadjacent when they differ in just one coordinate The graph H(d q) isdistance-transitive with intersection array

d(q-l)(d-l)(q-l)(q-l) 12 d

The eigenvalues are d(q - 1) - qi i = 0 1 d with multiplicities(d(q mdash 1) The intersection array determines the Hamming graphH(d q) uniquely except when q = 4 in that case there are other graphswith the same intersection array [BCN p 262]


21b The spectrum of Ok The eigenvalues of the odd graph Ok arei = (-lY(k-i) (0 lt i lt femdash 1) and

i) =m

21c Elementary conditions on the intersection array The followingconditions must be satisfied by the intersection array of any distance-regular graph Proofs may be found in Biggs (1976)(1) If a = 0 and a^^Q then 02 gt C2(2) If ax = 1 theno2 gt c2(3) If a-i mdash 2 and fc is not a multiple of 3 then c2 gt 2

21d Integrality of all intersection numbers Since the matrices Bi arethe images of the A under a faithful representation it follows that theysatisfy the relation B = u(B) (0 lt i lt d) Since (Bh)j is the numberShij it follows that each of the matrices computed by means of thisformula must have integral entries

21e The Krein conditions Define

n(0 lt i lt d)

where the L4 are as in the proof of Theorem 214 The E are mutuallyorthogonal idempotent and form a basis for the adjacency algebraThis algebra is closed under the pointwise product o of matrices becauseAi o Aj mdash SijAj It follows that there are real numbers qhij such that

Scott (1973) observed that these Krein parameters must be non-negativeThus we have a new set of feasibility conditions which can be statedexplicitly as follows

=E-(r = 0

21f An array which is not realisable The array 98 14 is feasiblein the sense of Definition 215 We have

and the eigenvalues are 91 mdash5 with multiplicities 1216 respectivelyThe conditions given in 21c are satisfied and also 21d since

B mdash090

108

045

B2 = laquoa(B) =0018

0810

1512


However the Krein condition g222 gt 0 in the notation of 21e does nothold An elementary proof that this array is not realisable was given byBiggs (1970)

21g Feasibility conditions for strongly regular graphs A strongly regu-lar graph as defined in 3c is a distance-regular graph with intersectionarray k k mdash a mdash 1 lc The eigenvalues and their multiplicities canbe computed by the elementary methods described in 3d or by thegeneral methods described in this chapter A good survey is given bySeidel (1979) In addition to the feasibility conditions which hold fordistance-regular graphs in general there is a useful absolute bound

n lt -m(m + 3)

where n is the number of vertices and m is the multiplicity of either oneof the eigenvalues A ^ k For example this test shows that the arrayconsidered in 21f is not realisable

21hThe friendship theorem If in a finite set of people each pair ofpeople has precisely one common friend then someone is everyonesfriend (Friendship is interpreted as a symmetric irreflexive relation)The result may be proved as follows Let F denote the graph whosevertices represent people and whose edges join friends Then F is eithera graph consisting of a number of triangles all with a common vertex or astrongly regular graph with intersection array kk mdash 211 The arrayis not feasible so the first possibility must hold This is an unpublishedproof of G Higman for other proofs see Hammersley (1981)

21i Distance-regular and distance-transitive graphs with degree 3 Biggsand Smith (1971) proved that there are exactly 12 distance-transitivegraphs with degree 3 They are (i) the symmetric cubic graphs withn lt 30 vertices listed in 18h with the exception of P(83) and P(125)(ii) the threefold covering of 5(3) with n = 90 vertices described in 19c(iii) the expansion of H with n = 102 vertices described in 18g

Biggs Boshier and Shawe-Taylor (1986) showed that in the distance-regular case there is just one other graph which has 126 vertices (see23b)

21j Perfect codes in distance-regular graphs The definition of a per-fect e-code in a graph was given in 3k Let Vi(X) be the polynomialsassociated with a distance-regular graph F and let

Xi(X) = laquoo(A) + laquo i (A) + + Vi(X) (0ltilt d)


If there is a perfect e-code in F then xe(A) is a factor of xlti(A) in the ringof polynomials with rational coefficients This implies that the zeros ofxe(X) must be eigenvalues of F This result was first established by SPLloyd in the classical case of a cube or Hamming graph Biggs (1973c)gave a proof for the general distance-transitive case and Delsarte (1973)proved similar results in a more general context

21k Sporadic groups and graphs Several of the sporadic simple groupscan be represented as the automorphism group of a distance-transitivegraph A typical example is the distance-transitive graph with 266 ver-tices which has degree 11 diameter 4 and intersection array 11106111511 The automorphism group of this graph is Jankos simplegroup of order 175 560 As usual the reader should consult [BCN] for afull account

211 The permutation character If F is a distance-transitive graph withdiameter d then the permutation character corresponding to the rep-resentation of Aut(F) on VT is the sum of d + 1 irreducible characters

X = 1 + Xi + bull bull bull + Xd

and the labelling can be chosen so that the degree of i is m(Ai) (0 lti lt d) This can be deduced from the results of Wielandt (1964) seealso [BCN p 137]

22

Imprimitivity

In this chapter we investigate the relationship between primitivity anddistance-transitivity We shall prove that the automorphism group of adistance-transitive graph can act imprimitively in only two ways bothof which have simple characterizations in terms of the structure of thegraph

We begin by summarizing some terminology If G is a group of per-mutations of a set X a block B is a subset of X such that B and g(B)are either disjoint or identical for each g in G If G is transitive on Xthen we say that the permutation group (X G) is primitive if the onlyblocks are the trivial blocks that is those with cardinality 0 1 or XIf B is a non-trivial block and G is transitive on X then each g(B) is ablock and the distinct blocks g(B) form a partition of X which we referto as a block system Further G acts transitively on these blocks

A graph F is said to be primitive or imprimitive according as thegroup G = Aut(F) acting on VT has the corresponding property Forexample the ladder graph L3 is imprimitive there is a block systemwith two blocks the vertices of the triangles in L3

Proposition 221 Let T be a connected graph for which the groupof automorphisms acts imprimitively and symmetrically (in the sense ofDefinition 155) Then a block system for the action of Aut(F) on VTmust be a colour-partition ofT

Proof Suppose that VT is partitioned by the block system


Then we may select one block call it C and elements g^ in Aut(r)such that

poundlaquogt = gMc (1 lt i lt I)

Suppose C contains two adjacent vertices u and v Since F is symmetricfor each vertex w adjacent to u there is an automorphism g such thatg(u) = u and g(v) mdash w Then u belongs to C n lt7(C) and C is a blockso C = g(C) and w belongs to C Since w was any vertex adjacent to vthe set Fi (u) is contained in C and by repeating the argument we canshow that F2(u)T3(u) are contained in C Since F is connected wehave C = VT This contradicts the hypothesis of imprimitivity and soour assumption that C contains a pair of adjacent vertices is false ThusC is a colour-class and since each block B^ is the image of C under anautomorphism the block system is a colour-partition bull

This result is false if we assume only that the graph is vertex-transitiverather than symmetric The ladder graph L3 mentioned above providesa counter-example

The rest of this chapter is devoted to an investigation of the relation-ship between primitivity and distance-transitivity We shall show thatin an imprimitive distance-transitive graph the vertex-colouring inducedby a block system is either a 2-colouring or a colouring of another quitespecific kind

Lemma 222 Let F be a distance-transitive graph with diameter dand suppose B is a block for the action oAut(F) on VT If B containstwo vertices u and v such that d(uv) = j (1 lt j lt d) then B containsall the sets rrj(u) where r is an integer satisfying 0 lt rj lt d

Proof Let w be any vertex in Tj(u) Since F is distance-transitivethere is an automorphism g such that g(u) mdash u and g(v) = w Thus uis in B fl g(B) and since B is a block B = g(B) and w is in B So

rraquo c BIf z is in F2j(u) there is a vertex y e Tj(u) for which d(yz) = j

Since d(zy) = d(uy) and both u and y are in B it follows by arepetition of the argument in the previous paragraph that z is in Band so F2j(u) C B Further repetitions of the argument show thatTTj (u) C B for each r such that rj ltd bull

For the rest of this chapter we use the symbol d to denote the largesteven integer not exceeding d

Imprimitivity 175

Proposition 223 Let F be a distance-transitive graph with diameterd and degree k gt 3 Then a non-trivial block for the action of Aut(F)on VT which contains the vertex u must be one of the following sets

Ba(u) = uurd(u) Bb(u) = uur2(laquo)ur4(laquo)uurv(u)Proof Suppose B is a non-trivial block containing u and is not theset Ba(u) Then B contains a vertex v ^ u such that d(u v) mdash j lt dand consequently Tj(u) C B

Consider the numbers Cjajbj in the intersection array of F Wemust have aj = 0 because if a were non-zero then B would containtwo adjacent vertices which is impossible by Proposition 221 Since

cj + a-j +bj = k gtZ

one of Cjbj is at least 2 Prom parts (2) and (3) of Proposition 204it follows that one of Cj+bj-i is at least 2 and consequently Tj(u)contains a pair of vertices at distance 2 Thus B contains the set Bb(u)If it contained any other vertices it would contain two adjacent verticesand would be the trivial block VT We deduce that B mdash Bb(u) asrequired bull

The cube Qz is an example of an imprimitive distance-transitive graphwith diameter d mdash 3 so d mdash 2 here One block system consists of foursets of the form u U ^ ( u ) of size two while another block systemconsists of two sets of the form u U F2(u) of size four This exampleillustrates the fact that both types of imprimitivity allowed by Proposi-tion 223 can occur in the same graph

Another instructive example is the cocktail-party graph CP(s) asdefined on p 17 Here there are s blocks u U ^ (u ) each of size twoand since d = d = 2 these blocks are simultaneously of type Ba (u) andBb(u) The next lemma clears up this case

Lemma 224 Let T be a distance-transitive graph with girth 3 anddiameter d gt 2 in which the set

Bb(u) =laquour2(ti)UU Td (u)

is a block Then d mdash 2 and consequently Bb(u) = u U ^ ( u ) = Ba(u)Proof Since F contains triangles and is distance-transitive every or-dered pair of adjacent vertices belongs to a triangle Choose adjacentvertices v euro Tu) V2 euro F2(u) then there is some vertex z such thatvv2z is a triangle If z were in F2(u) then 2amp(u) would contain adjacentvertices contrary to Proposition 221 Thus z must be in Fi(w)

If d gt 3 we can find a vertex v3 6 F3(M) which is adjacent to v2


Figure 18 illustrating the proof of Lemma 224

(Figure 18) But then ^(13) contains the adjacent vertices v and zand if h is an automorphism of F taking u to v3 h(Bb(u)) is a blockcontaining adjacent vertices again contradicting Proposition 221 Thuswe must have d = 2 bull

Proposition 225 Let T be a distance-transitive graph with diameterdgt3 and degree k gt 3 Then

x - Bb(u) = u u r2(u) u u vd- (u)is a block if and only ifTis bipartite

Proof Suppose F is bipartite If X is not a block then there isan automorphism g of F such that X and g(X) intersect but are notidentical This would imply that there are vertices x and y in X forwhich g(x) euro X but g(y) pound X so that d(xy) is even and d(g(x)g(y))is odd From this contradiction we conclude that X is a block

Conversely suppose X is a block A minimal odd cycle in F has length2j + 1 greater than 3 by Lemma 224 We may suppose this cycle to beuu W1V1V2W2 bull bull U2U where

uiu-2 e Ti(u) wiw2 euro Fj_i(w) viv2 6 Tj(u)

and if j = 2 then laquoi = w and u2 mdash w2 If j is even then X containsthe adjacent vertices vi and v2 and so X = VT a contradiction Ifj is odd we have for i mdash 12 d(uWi) mdash d(uiVi) and so there is anautomorphism hi taking u to u and Wi to w Thus Yt mdash hiX)samp blockcontaining Ui and Uj But since F contains no triangles d(uiu2) = 2and so u2 pound Vi Consequently Fi = Y2 and we have adjacent verticesî^2 in Y so that Yx = FFX = VF From this contradiction itfollows that F has no odd cycles and is bipartite bull

Lemma 224 and Proposition 225 lead to the conclusion that if ablock of the type Bb(u) exists in a distance-transitive graph F theneither d = 2 in which case the block is also of type Ba(u) or d gt 3 andF is bipartite The complete tripartite graphs KTTr are examples of thefirst case and are clearly not bipartite

Imprimitivity 177

We shall now show that graphs which have blocks of type Ba (u) canalso be given a simple graph-theoretical characterization

Definition 226 A graph of diameter d is said to be antipodal iffor any vertices u v w such that d(u v) = d(u w) = d it follows thatd(v w) = d or v = w

The cubes Qk are trivially antipodal since every vertex has a uniquevertex at maximum distance from it these graphs are at the same timebipartite The dodecahedron is also trivially antipodal but it is notbipartite Examples of graphs which are non-trivially antipodal and notbipartite are the complete tripartite graphs Krgtrir which have diameter2 and the line graph of Petersens graph which has diameter 3

Proposition 227 A distance-transitive graph F of diameter d has ablock Ba(u) = u U rlti(u) if and only ifT is antipodal

Proof Suppose F is antipodal Then if x is in Ba(u) it follows thatBa(u) = x U Td(x) = Ba(x) Consequently if g is any automorphismof F and z is in Ba(u) rg(Ba(u)) then

Ba(u) = zuTd(z)=g(Ba(u))

so that Bau) is a blockConversely suppose Ba(u) is a block and vw belong to Fû) (v ^

w) Let dvw) = j (1 lt j lt d) and let h be any automorphismof F such that h(v) mdash u Then h(w) is in Fj(w) Also h(w) belongsto h(Ba(u)) mdash Ba(u) since hBa(u)) intersects Ba(u) (u is in bothsets) and Ba(u) is a block This is impossible for 1 lt j lt d so thatd(v w) mdash d and F is antipodal

Theorem 228 (Smith 1971) An imprimitive distance-transitive graphwith degree k gt 3 is either bipartite or antipodal (Both possibilities canoccur in the same graph)

Proof A non-trivial block is either of the type Ba(u) or Bbu) In thecase of a block of type Bb(u) Proposition 225 tells us that either thegraph is bipartite or its diameter is less than 3 If the diameter is 1then the graph is complete and consequently primitive If the diameteris 2 a block of type Bb(u) is also of type Ba(u) Consequently if thegraph is not bipartite it must be antipodal bull

The notion of primitivity can be defined without reference to a groupaction in the following way Given a graph F with diameter d letFj (1 lt i lt d) be the graph whose vertices are the same as those of Ftwo vertices being adjacent in F if and only if they are at distance i in F


Then F is said to be imprimitive if any of the graphs Fi is disconnectedIt is easy to see that for a bipartite graph F2 has two componentsand for an antipodal graph F^ is the disjoint union of complete graphsUsing this definition Smiths theorem and its proof can be extended todistance-regular graphs (see [BCN p 140])

The complete graphs are primitive and distance-transitive Otherfamilies with the same properties are line graphs of a certain kind Apartfrom these families primitive distance-transitive graphs are scarce andwe give them a special name

Definition 229 An automorphic graph is a distance-transitive graphwhich is primitive and not a complete graph or a line graph

For instance of the 12 distance-transitive graphs with degree 3 (21i)only three are automorphic They are Petersens graph Coxeters graph(the expansion of Y with 28 vertices) and the expansion of H with 102vertices The odd graph O4 is the only automorphic graph with degree4 Many more details may be found in [BCN]

Additional Results

22a The derived graph of an antipodal graph Let F be a distance-transitive antipodal graph with degree k and diameter d gt 2 Definethe derived graph F by taking the vertices of F to be the blocks u UTd(u) in F two blocks being joined in F whenever they contain adjacentvertices of F Then F is a distance-transitive graph with degree k anddiameter equal to [d2j (Smith 1971)

22b The icosahedron and the dodecahedron The icosahedron and thedodecahedron D are distance-transitive with

i(I) = 521 125 i(D) = 32111 11123

Both graphs are antipodal and the derived graphs are K6 and O3

22c The intersection array of an antipodal covering We can look atthe construction in 22a from the opposite point of view as follows Adistance-regular graph F is an antipodal r-fold covering of the distance-regular graph F if F is antipodal its derived graph is F and |VF| =r|VT| It turns out that the intersection array of F is related to theintersection array k b 6^-1 1C2Cd of F in one of two waysEither (i) F has even diameter 2d gt 2 and

Imprimitivity 179

or (ii) F has odd diameter Id + 1 and for some positive integer t suchthat (r mdash l)t lt min(6ltj_ia(j) and cd lt t we have

t(f) = k amp ampd_i (r - ljicjcd-i bull bull c21

l c 2 Cdtbd-i bull bull bull bik

Clearly the total number of possibilities is finite and r lt k in any case

22d Antipodal coverings of Kkk Let F be a distance-regular graphwhich is an antipodal r-fold covering of Kkk- Then it follows from 22cthat r must divide k and if rt mdash k the intersection array for F is

kk - lk - tlltk - lk

This array is feasible (provided that r divides k) and the spectrum of F

S p e c F = ( ^ 2kdeg_1 fc^

In the case r = k the existence of F implies the existence of a projectiveplane of order k (Gardiner 1974)

22e Distance-regular graphs with diameter three A distance-regulargraph with diameter three is antipodal bipartite or primitive (in theextended sense defined on p 177) In the antipodal case the intersectionarray is of the form k (r mdash1)71 17 k and the graph is an antipodalr-fold covering of Kk+i This case has been the subject of several paperssee Biggs (1982c) Cameron (1991) Godsil and Hensel (1992) In thebipartite case the intersection array is of the form k kmdash lk-X 1 Xkand the graph is the incidence graph of a symmetric 2-design with pa-rameters (v k A) where v mdash k(k mdash 1)A+1 Several families of primitivegraphs are known and some sporadic ones [BCN pp 425-431]

22f An automorphic graph with k = 5 and d mdash 3 Let L = a b c d e and TV = 123456 The following table establishes a bijection be-tween the 15 single-transpositions on L and the 15 triple-transpositionson N

(O6)H

(ae) t-(bd)^(cd)^(de)K

+ (15)(23)(46)- (12)(36)(45)- (14)(25)(36)-gt (16)(23)(45)- (15)(26)(34)

(ac)i-

(laquo)-(ce) H

-gt (14)(26)(35)-raquo (16)(25)(34)^ (16)(24)(35)- (13)(25)(46)- (12)(35)(46)

(ad)v-(6c) H(bf)-(c)( e ) -

bull+ (13)(24)(56)-gt (12)(34)(56)- (13)(26)(45)- (15)(24)(36)- (14)(23)(56)

Define a graph F whose vertex-set is L x N and in which (hn) isadjacent to (hn2) if and only if the transposition (rin2) is one ofthose corresponding to (hfo)- Then F is an automorphic graph withdegree 5 and diameter 3 Its intersection array is 542 114 and itsautomorphism group is Aut S$

23

Minimal regular graphs with given girth

Results on the feasibility of intersection arrays can be applied to a widerange of combinatorial problems The last chapter of this book dealswith a graph-theoretical problem which has been the subject of muchresearch We shall study regular graphs whose degree (k gt 3) and girth(g gt 3) are given For all such values of k and g there is at least onegraph with these properties (Sachs 1963) and so it makes sense to ask forthe smallest one We note that when k = 2 the cycle graphs provide thecomplete answer to the problem and so we shall be concerned primarilywith the case k gt 3

Proposition 231 (1) The number of vertices in a graph with degreek and odd girth g mdash 2d + 1 is at least

no(kg) = l + k + k(k-l) + + k(k- l)^g-3l

If there is such a graph having exactly no(kg) vertices then it isdistance-regular with diameter d and its intersection array is

fcfc-lfc-lfc-lllll

(2) The number of vertices in a graph with degree k and even girth g = 2dis at least

no(kg) = l + kIf there is such a graph having exactly no(kg) vertices then it is bipar-tite and distance-regular with diameter d its intersection array is

k k ~ 1 k 1 K 1 1 1 1 1 kj

Minimal regular graphs with given girth 181

Proof (1) Suppose that F is a graph with degree k and girth g = 2d+land let (u v) be any pair of vertices such that d(u v) = j (1 lt j lt d)The number of vertices in Tjî(v) adjacent to u is 1 otherwise weshould have a cycle of length at most 2j lt 2d + 1 in F Using thestandard notation (Definition 205) we have shown the existence of thenumbers c = lCd = 1 Similarly if 1 lt j lt d then there areno vertices in Tj(v) adjacent to u otherwise we should have a cycle oflength at most 2j + l lt 2cf+l This means that aj = 0 and consequentlybj = k mdash aj mdash Cj = k mdash 1 for 1 lt j lt d It follows that the diameter ofF is at least d and that F has at least no(k g) vertices If F has justnokg) vertices its diameter must be precisely d which implies thataltj = 0 and F has the stated intersection array

(2) In this case the argument proceeds as in (1) except that cltj maybe greater than one Now the recurrence for the numbers ki = |Fj(v)|shows that kd is smallest when a = k if this is so then F has at leastno(kg) vertices If F has exactly no(fc g) vertices then its diameter isd and it has the stated intersection array The form of this array showsthat F has no odd cycles and so it is bipartite bull

Definition 232 A graph with degree k girth g and such that thereare no smaller graphs with the same degree and girth is called a (k g)-cage A (kg)-cage with no(kg) vertices is said to be a Moore graph ifg is odd and a generalized polygon graph if g is even (The reasons forthe apparently bizarre terminology are historical and may be found inthe references given below)

We have already remarked that a (k g)-cage exists for all k gt 3 andg gt 3 For example Petersens graph O3 is the unique (35)-cage ithas 10 vertices and no(35) = 10 so it is a Moore graph On the otherhand the unique (37)-cage has 24 vertices (see 23c) and no(37) = 22so there is no Moore graph in this case The main result of this chapteris that Moore graphs and generalized polygon graphs are very rare

In the cases g = 3 and g mdash 4 the intersection arrays in question are

fcl and fcfc-lljfc

and these are feasible for all k gt 3 It is very easy to see that eacharray has a unique realisation - the complete graph K^+i and the com-plete bipartite graph Kkik respectively Thus when g = 3 we have aunique Moore graph Kk+i and when g = 4 we have a unique generalizedpolygon graph Kkk-

When g gt 5 the problem is much more subtle both in the technicaldetails and in the nature of the solution The results are due to a number


of mathematicians The generalized polygon case was essentially solvedby Feit and Higman (1964) the Moore graph case was investigated byHoffman and Singleton (1960) Vijayan (1972) Damerell (1973) andBannai and Ito (1973)

We shall apply the algebraic techniques developed in Chapter 21 toboth cases in a uniform manner Specifically we investigate the feasi-bility of the intersection matrix

rok

10

J f c - 110- 1

10

fc-1c

k-cjwhich subsumes by putting c = 1 and c = k the intersection matricesof Moore graphs and generalized polygon graphs

Suppose that A is an eigenvalue of B and that the correspondingstandard left eigenvector is u(A) = [UQ(A) laquoI(A) ultj(A)] Then fromthe equations u(A)B = Au(A) and uo(A) = 1 we deduce that u(X) =Xk and

() cud-i(X) + (k - c - )ud) = 0

The equations () give a recursion which enables us to express Ui()as a polynomial of degree i in A for 0 lt i lt d The equation ()then becomes a polynomial equation of degree d + 1 in A In fact ()represents the condition that A is an eigenvalue it is the characteristicequation of B

Put q = yk mdash 1 and suppose that |A| lt 2q so that we may writeA = 2q cos a for some a 0 lt a lt ir (this assumption will be justified inthe course of the ensuing argument) The solution to the recursion ()can be found explicity

q2 sin(i + l )a mdash sin(i mdashkq1 sin a

(1 lt i lt d)

Lemma 233 With the above notation the number 2qcosa is aneigenvalue of B if and only if

c-Vqsm(d+ sin(d - l)a = 0

Proof The stated equation results from substituting the explicit formsof Ud-i and Ud in the equation () which is the characteristic equationof B bull


Proposition 234 (1) Let g = 2d and suppose F is a generalizedpolygon graph for the values (kg) Then F has d+l distinct eigenvalues

k-k 2qcosirjd (j = 12 d - 1)

(2) Let g = 2d + 1 and suppose F is a Moore graph for the values (k g)Then F has d+l distinct eigenvalues

k 2gcosa (j = 12 d)

where the numbers QJ otd o-re the distinct solutions in the interval0 lt a lt 7T of the equation qsin(d + l)a + sin da = 0Proof (1) The existence of the eigenvalues k and mdash k follows from thefact that F is fc-regular and bipartite Now the eigenvalues of F are (byProposition 212) the d + l eigenvalues of its intersection matrix whichis the matrix given above with c = k In that case A = 2qcosa is aneigenvalue of B if and only if

qsin(d + l)a + A sin da + qsin(d mdash l)a = 0This reduces to (2gcosa + k) sin da mdash 0 and since k2q gt 1 when k gt3 the only possibility is that sin da = 0 Thus in the range 0 lt a lt TTthere are d mdash 1 solutions a = irjd corresponding to j = 1 d mdash 1and we have the required total of d + 1 eigenvalues in all

(2) Since F is fc-regular k is an eigenvalue As in (1) we now seekeigenvalues A = 2gcosa of B this time with c = 1 The equation ofLemma 233 reduces to

A = q sin(d + l)a + sin da = 0

For 1 lt j lt d A is strictly positive at 6j = (j mdash ^)7r(d+1) and strictlynegative at cpj = (j + ^)n(d +1) Hence there is a zero aj of A in eachone of the d intervals ( 4gtj) Thus we have the required total of d + 1eigenvalues in all bull

We now have enough information to calculate the multiplicities of theeigenvalues and to test the feasibility of the corresponding intersectionarray Suppose that A is an eigenvalue of B The multiplicity of A asan eigenvalue of the putative graph is given by Theorem 214 m(A) =n(u(A) v(A)) We shall use this in the form m(A) = n^fcjU(A)2For our matrix B we have ko = 1 h = k(k - I)11 (1 lt i lt d mdash 1) andkd = c~lkk mdash I)1 Also for an eigenvalue A = 2qcosa we have

2

kq1 sin a= (2hksin2a)~1(E + Fcos2ia + Gsin2ia) (1 lt i lt d)

where we have written

h = q = fc mdash 1 E = (h +1) mdash 2icos2a


F = 2h-(h2 + l)cos2a G = (h2 - 1)sin2a

Allowing for the anomalous form of kd by means of a compensating termwe can sum the trigonometric series involved in ^kiUiX)2 and obtain

1 + (2hksin2 a ) - 1 dE + Fcos(d + l)a + Gsin(d + 1sin a

)

Fortunately this expression can be simplified considerably in the twocases c = 1 and c = k which are of particular interest

Proposition 235 IfA ^ plusmnfc is an eigenvalue of a generalized polygongraph with girth g = 2d then its multiplicity is given by

nk (4h-2

If X k is an eigenvalue of a Moore graph with girth g = 2d+ 1 thenits multiplicity is given by

) 7Proof In the case of even girth c = k and we know that A = 2q cos ais an eigenvalue if and only if sin da = 0 In this case the expression forJ2kiUi()2 becomes

1 + (2ifc sin2 a)-l[dE + hk1 (E + F) = (2hk sin2 a^dE

On putting 2d = g A = 2qcosa this leads to the formula givenIn the case of odd girth c = 1 and we know that A = 2gcosa is an

eigenvalue if and only if

qsin(d+ l)a + sin da = 0

From this equation we havemdashosina mdashosina

tan da = sin da =1 + q cos a

sin(d + l)a = mdash cos(d+l)a =V k + A Vfc + A

Substituting for the relevant quantities in the general expression andputting g = 2d + 1 we obtain after some algebraic manipulation thestated formula bull

We are now ready for the main theorem which is the result of thecombined efforts of the mathematicians mentioned earlier in this chapter


Theorem 236 The intersection array for a generalized polygon graphwith k gt 3 g gt 4 is feasible if and only if g euro 46812 The inter-section array for a Moore graph with k gt 3 g gt 5 is feasible if and onlyif g = 5 and kpound 3757Proof Suppose g is even g = Id Then a generalized polygon graphhas d - 1 eigenvalues Xj = 2q COS(TTjd) with multiplicities

m ( A j j ~ g k-Xj)-

If m(Aj) is a positive integer Af is rational which means that cos27rdis rational But it is well known (see for example Irrational Numbers byI Niven (Wiley 1956) p 37) that this is so if and only if d G 2346

The case when g is odd presents more problems We shall deal withg = 5 and g = 7 separately and then dispose of g gt 9 Suppose g = 5Then the characteristic equation

g sin 3a + sin 2a = 0

reduces in terms of A = 2qcosa to A2 + A mdash (k mdash 1) = 0 Thus thereare two eigenvalues Ai = |(mdash1 + gtD) and A2 = mdash1 mdash VAD) whereZ = 4fc mdash 3 We have n = 1 + k2 and putting this in the formula form(A) we get

m _ (fc + fc3)(4fc-4-A2)W (Jfe-A)(6-2 + 5A)

If v^D is irrational we multiply out the expression above substituteA = ( -1 plusmn V^D) and equate the coefficients of v^D This gives 5m + c mdash2 = k + k3 where m = m(Ai) = m(A2) But there are three eigenvaluesin all k Ai A2 with multiplicities 1mm hence 1 + 2m = n = 1 + k2Thus 5fc2 mdash 4 = 2k3 which has no solution for A gt 3 Consequently VDmust be rational s = y~D say Then k = (s2 +3) and substituting forAi and k in terms of s in the expression for mi = m(Ai) we obtain thefollowing polynomial equation in s

s5 + s4 + 6s3 - 2s2 + (9 - mi)s - 15 = 0

It follows that s must be a divisor of 15 and the possibilities are s =13515 giving k = 13557 The first possibility is clearly absurdbut the three others do lead to feasible intersection arrays

Suppose g = 7 Then the characteristic equation

q sin 4a + sin 3a = 0

reduces in terms of A = 2qcosa to A3 + A2 mdash 2(k-l)X-(k-l) = 0 Thisequation has no rational roots (and consequently no integral roots) sincewe may write it in the form k mdash 1 = A2(A + 1)(2A +1) and if any prime


divisor of 2 A +1 divides x = X or A +1 it must divide 2 A +1 mdash a = A +1 orA which is impossible So the roots Ai A2 A3 are all irrational and theirmultiplicities are all equal to m say Then 1 + 3m = n = l + k mdash k2 + k3whereas k + m(Xi + A2 + A3) = trA = 0 But Ai + A2 + A3 = mdash 1 hence

m = k = -(A3 - fc2 + fc)

which is impossible for k gt 3 Thus there are no Moore graphs when5 = 7

Suppose g gt 9 We obtain a contradiction here by proving first thatmdash 1 lt Ai + Altf lt 0 and then showing that all eigenvalues must in factbe integers (The argument just fails in the case fc = 3 g mdash 9 but thiscan be discarded by an explicit calculation of the multiplicities)

Let ai (1 lt i lt d) be the roots of

A = qsin(d + )a + sin da = 0

and set ugt = n(d+1) The proof of Proposition 234 showed that a liesbetween ui2 and 3w2 and these bounds can be improved by notingthat A is positive at w and negative at w(l + l2q) Thus w lt a i ltltJJ(1 + l2q) and

0 lt 2gcoso mdash 2qcosai lt 2gcosw - 2gcosw(l + l2g)

= 2gcosw(l - cosuj2q) + 2qsinojsm(uj2q)

lt2qx - ( 2

In a similar way it can be shown that dw lt ad lt ugt(d + l2g) and

0 lt 2q cos duj mdash 2q cos ad lt us2

Adding the two inequalities and noting that

Ai = 2qcosa d = 2qcosad cosdu =mdashcosui

we have

-9w24 lt Ai + Xd lt 0

Now w2 = r2(d + I)2 lt TT252 lt 49 so - 1 lt Ai + Xd lt 0 as

promisedTo show that the eigenvalues must be integers we note first that since

the characteristic equation is monic with integer coefficients the eigen-values are algebraic integers The formula for m(X) is the quotient oftwo quadratic expressions in A and so m(X) is integral only if A is atworst a quadratic irrational Suppose A is a quadratic irrational Then

R(X) = gm(X)nk = (4ft - X2)(k - X)(f + A)


is rational number and this equation can be written in the form

(R(X) - 1)A2 + R(X)(f - k)X - (R(X)fk - Ah) = 0

But this must be a multiple of the minimal equation for A which ismonic with integer coefficients In particular

( - k)RX) Ah - A fk-AhW h e r e ~ -fc

must be an integer However = k + (k mdash 2)g gt fc so t gt k2 mdashAh)(f mdash k) = lt(fc mdash 2) and consequently |A mdash t gt g(k mdash 2) mdash k since|A| lt k Thus

for all k gt 3 g gt 9 (except when A = 3 = 9 as we have alreadynoted) Since S(X) is to be an integer we must have S(X) = 0 whichleads to the absurdity R(X) = m(X) = 0 Thus all eigenvalues A must beintegers which is incompatible with the inequality mdash 1 lt Ai + Aj lt 0and consequently disposes of all cases with g gt 9 bull

The question of the existence of graphs allowed by Theorem 236 is adifficult one and it contains some celebrated unsolved problems In thecase of even girth g = 2d we can relate the problem to existence of astructure known as a generalized d-gon defined as follows

Let (P L I) be an incidence system consisting of two disjoint finitesets P (points) and L (lines) and an incidence relation I between pointsand lines A sequence whose terms are alternately points and lines eachterm being incident with its successor is called a chain it is a properchain if there are no repeated terms except possibly when the firstand last terms are identical (when we speak of a closed chain) A (non-degenerate) generalized d-gon is an incidence system with the properties(a) each pair of elements of P U L is joined by a chain of length at mostd (b) there is a pair of elements of P U L for which there is no properchain of length less than d joining them (c) there are no closed chainsof length less than 2d

Denote by Gds t) a generalized d-gon with s points on each line andt lines through each point Given a Gd(k fc) the graph whose vertex-setis P U L and whose edge-set consists of incident pairs is a (fc 2d)-cagewith no(fc 2d) vertices The converse is also true Thus our generalizedpolygon graphs are just the incidence graphs of generalized d-gons withs = t


It is easy to construct a G2kk) for all k gt 2 the correspondinggraph is the complete bipartite graph Kk^ A G$(kk) is simply aprojective plane with k points on each line So the existence problemfor generalized polygon graphs of girth 6 is covered by the known resultson projective planes a fact noted by Singleton (1966) There is at leastone such plane whenever k mdash 1 is a prime power and none are known forwhich k mdash 1 is not a prime power Generalized quadrangles Gplusmnk k) arealso known to exist for all prime power values of k mdash 1 and generalizedhexagons Gsectk k) exist whenever k - 1 is an odd power of 3 Benson(1966) was the first to construct the graphs corresponding to the thelast two cases

In the case of odd girth g gt 3 the only Moore graphs allowed byTheorem 236 are those with g = 5 and k euro 3757 The graph withk = 3 is Petersens graph The graph with k = 7 was constructed andproved unique by Hoffman and Singleton (1960) a construction is givenin 23d The existence of a graph with k = 57 remains an enigma theresults of Aschbacher (1971) show that such a graph cannot be distance-transitive and so the construction if there is one is certain to be verycomplicated

Additional Results

23a Moore graphs and generalized polygon graphs with degree 3 In thecase k = 3 the Moore graphs of girth 3 and girth 5 (K4 and O3) exist andare unique There are no other Moore graphs of degree 3 by Theorem236 The generalized polygon graphs of girth 4 6 8 and 12 exist andare unique They are ^33 Heawoods graph 5(7) Tuttes graph Cl andthe incidence graph of the unique generalized hexagon with 63 pointsand 63 lines (see 23b)

23b The (312)-cage A direct construction of the generalized hexagongraph of degree 3 is as follows Given a unitary polarity of the projectiveplane PG(232) there are 63 points of the plane which do not lie ontheir polar lines and they form 63 self-polar triangles (Edge 1963) The(312)-cage is the graph whose 126 vertices are these 63 points and 63triangles with adjacent vertices corresponding to an incident (pointtriangle) pair

This graph is not vertex-transitive since there is no automorphismtaking a point vertex to a triangle vertex However it follows fromProposition 231 that it is distance-regular


23c Cages with degree 3 and g lt 12 All cases except g = 7910 and 11have been covered above In these cases we know from the general theorythat a (3 lt7)-cage must have more than no(3 g) vertices The (37)-cageis a graph with 24 vertices and it is unique details are given by Tutte(1966) There are numerous (39)-cages they have 58 vertices and thefirst one was found by Biggs and Hoare (1980) The fact that no smallergraph has degree 3 and girth 9 is the result of a computer search byB McKay There are three (310)-cages they have 70 vertices (OKeefeand Wong 1980) The size of the (3 ll)-cage is as yet unknown Sinceit is not a Moore graph it must have at least 96 vertices the smallestknown graph with degree 3 and girth 11 has 112 vertices

23d The Hoffman-Singleton graph The unique (75)-cage may be con-structed by extending the graph described in 22f as follows Add 14 newvertices called L N a b c d e f 123456 join L to a b c d e and N join N to 123456 and L Also join the vertex denoted by(ln) in 22f to I and n The automorphism group of this graph is thegroup of order 252 000 obtained from PSU(352) by adjoining the fieldautomorphism of GF(b2) (Hoffman and Singleton 1960)

23e Cages of girth 5 with 4 lt k lt 6 In these cases we know that a cageis not a Moore graph There is a unique (45)-cage with 19 vertices dueto Robertson (1964) There are several (55)-cages having 30 verticessee [BCN p 210] There is a unique (65)-cage (OKeefe and Wong1979) it has 40 vertices and it is the induced subgraph obtained bydeleting the vertices of a Petersen graph from the Hoffman-Singletongraph

23f Cages of girth 6 Recall (4d) that the excess of a fc-regular graphwith n vertices and girth g is e = n mdash no(kg) Biggs and Ito (1980)showed that for small values of e a fc-regular graph with girth 6 andexcess e = 2(77 mdash 1) is an 77-fold covering of the incidence graph of asymmetric (ufc77)-design

When 77 = 1 such a design is a projective plane and we have thegeneralized polygon graph as discussed above When 77 = 2 such adesign is called a biplane In this case it can be shown that a necessarycondition for the existence of a graph is that either k or k mdash 2 must bea perfect square (see Biggs 1981b) Such graphs with k = 3 and k = Ado exist but they they are not (k 6)-cages because for these values of kthere is a generalized polygon graph The first significant case is k = 11because here it is now known that there is no projective plane so the


graph (if it exists) would be an (116)-cage There are several biplanesbut the existence of a 2-fold covering has not been settled

When 77 = 3 coverings have been constructed for k = 47 and 12 Thecase k = 7 is particularly important because there is no projective planeor biplane in this case and so the graph is a (76)-cage (see OKeefe andWong (1981) and Ito (1981)) This is the last of the known cages

23g Families of graphs with large girth Graphs with small excess arevery special and we therefore adopt a wider definition of what is inter-esting in this context Let Fr be a family of fc-regular graphs suchthat Fr has nr vertices and girth gr We say that the family has largegirth if nr and gr both tend to infinity as r mdashbull 00 in such a way that

lim mdash - is a finite constant cr-oo gr

It follows from the explicit form of no(k g) that c cannot be less than 05For many years the existence of families with large girth was establishedonly by non-constructive means these arguments showed that there arefamilies with c = 1 Weiss (1984) showed that in the case k = 3 thefamily of sextet graphs S(p) defined in 18b has c = 075 and LubotzkyPhillips and Sarnak (see 23h) constructed families which attain the samevalue for infinitely many values of k A simple construction for cubicgraphs with large girth (but with c gt 1) was given by Biggs (1987)

23h The graphs of Lubotzky Phillips and Sarnak Let p be a primecongruent to 1 modulo 4 and let H denote the set of integral quaternionsa = (00010203) Define A(2) to be the set of it-equivalence classesof elements a of H with a = 1 mod 2 and ||a|| a power of p whereaR(3 if plusmnpra = psf3 Denote by 5 the set of elements of H satisfying||a|| = p a = 1 mod 2 and ao gt 0 There are (p + l)2 conjugatepairs a a in S and the Cayley graph of A(2) with respect to S is theinfinite (p + l)-regular tree

Now let q be another prime congruent to 1 modulo 4 such that q gtyp and (p I q) = mdash1 Denote by A(2g) the normal subgroup of A(2)consisting of those classes represented by a with at 0203 divisible by 2qThe Cayley graph of SA(2q) with respect to A(2)A(2o) is a bipartite(p + l)-regular graph with qq2 mdash 1) vertices and girth approximately41ogpg For further details see Lubotzky Phillips and Sarnak (1988)Biggs and Boshier (1990)

REFERENCES

Adam A (1967) Research problem 2-10 JCT 2 393 [126]Adelson-Velskii GM et al (1969) Example of a graph without atransitive automorphism group [in Russian] Soviet Math Dokl 10440-441 [162]Alon N and VD Milman (1985) Ai isoperimetric inequalities forgraphs and superconcentrators JCT(B) 38 73-88 [2258]Alspach B D Marusic and L Nowitz (1993) Constructing graphswhich are 12-transitive J Austral Math Soc to appear [120]Anthony MHG (1990) Computing chromatic polynomials Ars Com-binatoria 29C 216-220 [Ill]Aschbacher M (1971) The non-existence of rank three permutationgroups of degree 3250 and subdegree 57 J Algebra 19 538-540 [188]Austin TL (1960) The enumeration of point labelled chromatic graphsand trees Canad J Math 12 535-545 [41]Babai L (1979) Spectra of Cayley graphs JCT(B) 27 180-189 [128]Babai L (1981) On the abstract group of automorphisms Combina-torics (ed HN Temperley Cambridge UP) 1-40 [117]Baker GA (1971) Linked-cluster expansion for the graph-vertex col-oration problem JCT(B) 10 217-231 [818693]Bannai E and T Ito (1973) On Moore graphs J Fac Sci UnivTokyo Sec 1A 20 191-208 [182]Benson CT (1966) Minimal regular graphs of girth eight and twelveCanad J Math 18 1091-1094 [188]

192 References

Beraha S and J Kahane (1979) Is the four-color conjecture almostfalse JCT(B) 27 1-12 [71]Beraha S J Kahane and NJ Weiss (1980) Limits of chromatic zerosof some families of maps JCT(B) 28 52-65 [7172]Biggs NL (1970) Finite Groups of Automorphisms (Cambridge Uni-versity Press) [171]Biggs NL (1973a) Expansions of the chromatic polynomial DiscreteMath 6 105-113 [81]Biggs NL (1973b) Three remarkable graphs Canadian J Math 25397-411 [103111]Biggs NL (1973c) Perfect codes in graphs JCT(B) 15 289-296 [172]Biggs NL (1976) Automorphic graphs and the Krein condition GeomDedicata 5 117-127 [170]Biggs NL (1977a) Colouring square lattice graphs Bull LondonMath Soc 9 54-56 [96]Biggs NL (1977b) Interaction Models (Cambridge UP) [ 8087110]Biggs NL (1978) On cluster expansions in graph theory and physicsQuart J Math (Oxford) 29 159-173 [818790]Biggs NL (1979) Some odd graph theory Ann New York Acad Sci319 71-81 [58137]Biggs NL (1980) Girth valency and excess Lin Alg Appl 3155-59 [28]Biggs NL (1981a) Aspects of symmetry in graphs Algebraic Methodsin Graph Theory ed L Lovasz and V Sos (North-Holland) 27-35 [137]Biggs NL (1981b) Covering biplanes Theory and Applications ofGraphs ed G Chartrand (Wiley) 73-82 [189]Biggs NL (1982a) A new 5-arc-transitive cubic graph J Graph The-ory 6 447-451 [154]Biggs NL (1982b) Constructing 5-arc-transitive cubic graphs J Lon-don Math Soc (2) 26 193-200 [153]Biggs NL (1982c) Distance-regular graphs with diameter three AnnDiscrete Math 15 68-90 [179]Biggs NL (1984a) Presentations for cubic graphs ComputationalGroup Theory ed MD Atkinson (Academic Press) 57-63 [146]Biggs NL (1984b) Homological coverings of graphs J London MathSoc (2) 30 1-14 [154]Biggs NL (1987) Graphs with large girth Ars Combinatoria 25C73-80 [190]Biggs NL and AG Boshier (1990) Note on the girth of Ramanujangraphs JCT(B) 49 190-194 [190]

References 193

Biggs NL AG Boshier and J Shawe-Taylor (1986) Cubic distance-regular graphs J London Math Soc (2) 33 385-394 [163171]Biggs NL GR Brightwell and D Tsoubelis (1992) Theoretical andpractical studies of a competitive learning process Network 3 285-301

[58]Biggs NL RM Damerell and DA Sands (1972) Recursive familiesof graphs JCT(B) 12 123-131 [42 69 70 103]Biggs NL and MJ Hoare (1980) A trivalent graph with 58 verticesand girth 9 Discrete Math 30 299-301 [189]Biggs NL and MJ Hoare (1983) The sextet construction for cubicgraphs Combinatorica 3 153-165 [144152]Biggs NL and T Ito (1980) Graphs with even girth and small excessMath Proc Cambridge Philos Soc 88 1-10 [189]Biggs NL and GHJ Meredith (1976) Approximations for chromaticpolynomials JCT(B) 20 5-19 [96]Biggs NL and DH Smith (1971) On trivalent graphs Bull LondonMath Soc 3 155-158 [171]Birkhoff GD (1912) A determinant formula for the number of waysof coloring a map Ann Math 14 42-46 [76]Blokhuis A and AE Brouwer (1993) Locally K3t3 or Petersen graphsDiscrete Math 1067 53-60 [129]Bollobas B (1985) Random Graphs (Academic Press) [119]Bondy JA (1991) A graph reconstructors manual Surveys in Com-binatorics ed AD Keedwell (Cambridge UP) 221-252 [51]Bose RC and DM Mesner (1959) Linear associative algebras AnnMath Statist 30 21-38 [161]Bouwer IZ (1970) Vertex- and edge-transitive but not 1-transitivegraphs Canad Math Bull 13 231-237 [120]Bouwer IZ (1972) On edge- but not vertex-transitive regular graphsJCT(B) 12 32-40 [119]Bouwer IZ (1988) The Foster Census (Charles Babbage ResearchCentre Winnipeg) [147]Brylawski T (1971) A combinatorial model for series-parallel net-works Trans Amer Math Soc 154 1-22 [109]Cameron PJ (1980) 6-transitive graphs JCT(B) 28 168-179 [162]Cameron PJ (1982) There are only finitely many distance-transitivegraphs of given valency greater than two Combinatorica 2 9-13 [163]Cameron PJ (1991) Covers of graphs and EGQs Discrete Math 9783-92 [179]

194 References

Cameron PJ JM Goethals JJ Seidel EE Shult (1976) Linegraphs root systems and elliptic geometry J Algebra 43 305-327

[1822]Cayley A (1878) On theory of groups Proc London Math Soc 9126-133 [122]Cayley A (1889) A theorem on trees Quart J Math 23 376-378Collected papers 13 2628) [39]Chang S (1959) The uniqueness and nonuniqueness of the triangularassociation scheme Sci Record 3 604-613 [21]Chung F-K (1989) Diameters and eigenvalues J Amer Math Soc2 187-196 [22]Collatz L and U Sinogowitz (1957) Spektren endlicher Grafen AbhMath Sem Univ Hamburg 21 63-77 [12]Conder M (1987) A new 5-arc-transitive cubic graph J Graph Theory11 303-307 [148]Conder M and P Lorimer (1989) Automorphism groups of symmetricgraphs of valency 3 JCT(B) 47 60-72 [128146]Coulson CA and GS Rushbrooke (1940) Note on the method ofmolecular orbitals Proc Camb Philos Soc 36 193-200 [11]Coxeter HSM (1950) Self-dual configurations and regular graphsBull Amer Math Soc 59 413-455 [153154]Crapo HH (1967) A higher invariant for matroids JCT 2 406-417

[107]Crapo HH (1969) The Tutte polynomial Aeq Math 3 211-229

[102104]Cvetkovic DM (1972) Chromatic number and the spectrum of agraph Publ Inst Math (Beograd) 14 25-38 [57]Cvetkovic DM and P Rowlinson (1988) Further properties of graphangles Scientia A Math Sci 1 41-51 [51]Damerell RM (1973) On Moore graphs Proc Cambridge PhilosSoc 74 227-236 [160182]Delsarte P (1973) An algebraic approach to the association schemesof coding theory Philips Research Reports Suppl 10 [172]Djokovic DZ (1972) On regular graphs II JCT(B) 12 252-259 [140]Djokovic DZ and GL Miller (1980) Regular groups of automor-phisms of cubic graphs JCT(B) 29 195-230 [146]Edge WL (1963) A second note on the simple group of order 6048Proc Cambridge Philos Soc 59 1-9 [188]Egervary E (1931) Matrizok kombinatorius tulajdonsagairol MatFiz Lapok 38 16-28 [35]

References 195

Elspas B and JTurner (1970) Graphs with circulant adjacency ma-trices JCT 9 297-307 [126]Essam JW (1971) Graph theory and statistical physics DiscreteMath 1 83-112 [107]Feit W and G Higman (1964) The non-existence of certain generalizedpolygons J Algebra 1 114-131 [182]Frucht R (1938) Herstellung von Graphen mit vorgegebener abstrakterGruppe Compositio Math 6 239-250 [117]Frucht R (1949) Graphs of degree three with a given abstract groupCanad J Math 1 365-378 [117]Frucht R JE Graver and ME Watkins (1971) The groups of thegeneralized Petersen graphs Proc Cambridge Philos Soc 70211-218

[119]Gardiner AD (1973) Arc transitivity in graphs Quart J MathOxford (2) 24 399-407 [137]Gardiner AD (1974) Antipodal covering graphs JCT(B) 16 255-273

[179]Gardiner AD (1976) Homogeneous graphs JCT(B) 20 94-102 [120]Godsil CD (1981) GRRs for solvable groups Algebraic Methods inGraph Theory ed L Lovasz and V Sos (North-Holland) 221-239 [128]Godsil CD and AD Hensel (1992) Distance-regular covers of thecomplete graph JCT(B) 56 205-238 [179]Godsil CD and B Mohar (1988) Walk-generating functions and spec-tral measures of infinite graphs Lin Alg Appl 107 191-206 [13]Goldschmidt D (1980) Automorphisms of trivalent graphs Ann ofMath I l l 377-406 [143]Grimmett GR (1976) An upper bound for the number of spanningtrees of a graph Discrete Math 16 323-324 [42]Grimmett GR (1978) The rank polynomials of large random latticesJ London Math Soc 18 567-575 [94]Grone R and R Merris (1988) A bound for the complexity of a simplegraph Discrete Math 69 97-99 [42]Gross JL (1974) Voltage graphs Discrete Math 9 239-246 [149]Grotschel M L Lovasz and A Schrijver (1988) Geometric Algorithmsand Combinatorial Optimization (Springer New York) [35]Haemers W (1979) Eigenvalue Techniques in Design and Graph The-ory (Math Centrum Amsterdam) [57]Haggard G (1976) The chromatic polynomial of the dodecahedronTechnical Report No 11 (University of Maine) [69]

196 References

Hall JI (1980) Locally Petersen graphs J Graph Theory 4 173-187[129]

Hammersley J (1981) The friendship theorem and the love problemSurveys in Combinatorics ed EK Lloyd (Cambridge University Press)31-54 [171]Harary F (1962) The determinant of the adjacency matrix of a graphSIAM Review 4 202-210 [44]Heller I and CB Tompkins An extension of a theorem of DantzigsLinear Inequalities and Related Systems ed HW Kuhn andAW Tucker (Princeton UP) [3435]Heron A (1972) Matroid polynomials Combinatorics edDJA Welsh and DR Woodall (Institute of Mathematics and its Ap-plications) 164-202 [109]Hetzel D (1976) Uber regulare graphische Darstellung von auflosbarengruppen Diplomarbeit Technische Universitdt Berlin [128]Higman DG (1967) Intersection matrices for finite permutationgroups J Algebra 6 22-42 [161]Higman DG and CC Sims (1968) A simple group of order 44352000Math Zeitschr 105 110-113 [161]Hoffman AJ (1960) On the exceptional case in the characterizationof the arcs of a complete graph IBM J Res Dev 4 487-496 [21]Hoffman AJ (1963) On the polynomial of a graph Amer MathMonthly 70 30-36 [15]Hoffman AJ (1969) The eigenvalues of the adjacency matrix of agraph Combinatorial Mathematics and its Applications (Univ of NCarolina Press Chapel Hill) 578-584 [17]Hoffman AJ (1970) On eigenvalues and colorings of graphs GraphTheory and its Applications (Academic Press New York) 79-92 [57]Hoffman AJ and JB Kruskal (1956) Integer boundary points of con-vex polyhedra Linear Inequalities and Related Systems ed HW Kuhnand AW Tucker (Princeton UP) 223-246 [35]Hoffman AJ and RR Singleton (1960) On Moore graphs with diam-eters 2 and 3 IBM J Res Dev 4 497-504 [182188189]Holt D (1981) A graph which is edge-transitive but not arc-transitiveJ Graph Theory 5 201-204 [120]Horton JD and IZ Bouwer (1991) Symmetric Y-graphs and H-graphs JCT(B) 53 114-129 [147]Ito T (1981) On a graph of OKeefe and Wong J Graph Theory 587-94 [190]Ito T (1982) Bipartite distance-regular graphs of valency three LinAlg Appl 46 195-213 [153]

References 197

Ivanov A A (1983) Bounding the diameter of a distance-regular graphSoviet Math Dokl 28 149-152 [161]Izbicki H (1960) Regulare Graphen beliebigen Grades mit vorgegebe-nen Eigenschaften Monatshefte fur Math 64 15-21 [121]Jaeger F (1988) On Tutte polynomials and cycles of planar graphsJCT(B) 44 129-146 [104]Jaeger F (1991) Even subgraph expansions for the flow polynomial ofcubic plane maps JCT(B) 52 259-273 [110]Jaeger F DL Vertigan and DJA Welsh (1990) The computationalcomplexity of the Jones and Tutte polynomials Math Proc CambPhilos Soc 108 35-53 [105111]Jordan C (1869) Sur les assemblages des lignes J f d Reine uAngew Math 70 185-190 [119]Kelly PJ (1957) A congruence theorem for trees Pacific J Math 7961-968 [50]Kim D and IG Enting (1979) The limit of chromatic polynomialsJCT(B) 26 327-336 [96]Kirchhoff G (1847) Uber die Auflosung der Gleichungen auf welcheman bei der Untersuchung der linearen Verteilung galvanischer Stromegefiirht wird Ann Phys Chem 72 497-508 [3136]Kocay WL (1981) On reconstructing spanning subgraphs Ars Com-binatoria 11 301-313 [50]Las Vergnas M (1978) Eulerian circuits of 4-valent graphs imbedded insurfaces Algebraic Methods in Graph Theory ed L Lovasz and V Sos(North-Holland Amsterdam) 451-477 [104]Las Vergnas M (1988) On the evaluation at (33) of the Tutte poly-nomial of a graph JCT(B) 45 367-372 [104]Li N-Z and EG Whitehead (1992) The chromatic uniqueness of WwDiscrete Math 104 197-199 [69]Lieb EH (1967) Exact solution of the problem of the entropy of two-dimensional ice Phys Rev Letters 18 692-694 [95]Lorimer P (1989) The construction of Tuttes 8-cage and the Condergraph J Graph Theory 13 553-557 [148]Lovasz L (1979) On the Shannon capacity of a graph IEEE TransInform Theory 25 1-7 [51]Lubotzky A R Phillips and P Sarnak (1988) Ramanujan graphsCombinatorica 8 261-277 [190]Meredith GHJ (1976) Triple-transitive graphs J London Math Soc(2) 13 249-257 [162]Mohar B (1991) Eigenvalues diameter and mean distance in graphsGraphs Comb 7 53-64 [22]

198 References

Moon J (1967) Enumerating labelled trees Graph Theory and Theo-retical Physics (Academic Press New York) 261-272 [41]Moon J (1970) Counting labelled trees (Canadian Math CongressMontreal) [38]Motzkin T and EG Straus (1965) Maxima for graphs and a newproof of a theorem of Turan Canad J Math 17 533-540 [59]Mowshowitz A (1969) The group of a graph whose adjacency matrixhas all distinct eigenvalues Proof Techniques in Graph Theory (Aca-demic Press New York) 109-110 [117]Nagle JF (1971) A new subgraph expansion for obtaining coloringpolynomials for graphs JCT(B) 10 42-59 [80]Nerode A and H Shank (1961) An algebraic proof of Kirchhoffsnetwork theorem Amer Math Monthly 68 244-247 [36]OKeefe M and PK Wong (1979) A smallest graph of girth 5 andvalency 6 JCT(B) 26 145-149 [189]OKeefe M and PK Wong (1980) A smallest graph of girth 10 andvalency 3 JCT(B) 29 91-105 [189]OKeefe M and PK Wong (1981) The smallest graph of girth 6 andvalency 7 J Graph Theory 5 79-85 [190]Parsons TD (1980) Circulant graph imbeddings JCT(B) 29 310-320 [126]Petersdorf M and H Sachs (1969) Spektrum und Automorphismen-gruppe eines Graphen Combinatorial Theory and its Applications III(North-Holland Amsterdam) 891-907 [117125]Poincare H (1901) Second complement a lanalysis situs Proc LondonMath Soc 32 277-308 [32]Ray N (1992) Tutte algebras of graphs and formal group theory ProcLondon Math Soc 65 23-45 [88]Ray N and C Wright (1992) Umbral interpolation and the addi-tioncontraction tree for graphs Discrete Math 103 67-74 [72]Read RC (1968) An introduction to chromatic polynomials CombTheory 4 52-71 [69108]Read RC (1990) Recent advances in chromatic polynomial theoryProc 5th Carib Conf on Graph Theory and Computing [71]Robertson N (1964) The smallest graph of girth 5 and valency 4 BullAmer Math Soc 70 824-825 [189]Ronse C (1978) On homogeneous graphs J London Math Soc (2)17 375-379 [120]Rowlinson P (1987) A deletion-contraction algorithm for the char-acteristic polynomial of a multigraph Proc Roy Soc Edin 105A153-166 [11]

References 199

Sachs H (1962) Uber selbstkomplementare Graphen Publ MathDebrecen 9 270-288 [20]Sachs H (1963) Regular graphs with given girth and restricted circuitsJ London Math Soc 38 423-429 [180]Sachs H (1964) Beziehungen zwischen den in einem Graphen enthal-tenen Kreisen und seinem charakterischen Polynom Publ Math De-brecen 11 119-134 [49]Sachs H (1967) Uber Teiler Faktoren und characterische Polynomevon Graphen II Wiss Z Techn Hosch Ilmenau 13 405-412 [19]Schmitt W (1993) Hopf algebra methods in graph theory Pure ApplAlgebra to appear [88]Schur I (1933) Zur Theorie der einfach transitiven Permutationsgrup-pen SB Preuss Akad Wiss 598-623 [161]Schwenk AJ (1973) Almost all trees are cospectral New Directionsin the Theory of Graphs (Academic Press New York) [49]Scott LL (1973) A condition on Higmans parameters Notices AmerMath Soc 20 A97 (701-20-45) [170]Sedlacek J (1970) On the skeletons of a graph or digraph Combi-natorial Structures and Applications (Gordon and Breach New York)387-391 [42]Seidel JJ (1968) Strongly regular graphs with ( mdash 110) adjacencymatrix having eigenvalue 3 Lin Alg Appl 1 281-298 [1821]Seidel JJ (1979) Strongly regular graphs Surveys in Combinatoricsed B Bollobas (Cambridge UP) 157-180 [171]Seymour PD (1981) Nowhere-zero 6-flows JCT(B) 30 130-135 [30]Sheehan J (1974) Smoothly embeddable subgraphs J London MathSoc (2) 9 212-218 [120]Shrikhande SS (1959) The uniqueness of the L2 association schemeAnn Math Stat 30 781-798 [21]Sims CC(1967)Graphs and finite permutation groups Math Zeitschr95 76-86 [141]Singleton RR (1966) On minimal graphs of maximum even girth JComb Theory 1 306-332 [188]Smith DH (1971) Primitive and imprirnitive graphs Quart J MathOxford (2) 22 551-557 [177178]Smith JH (1970) Some properties of the spectrum of a graph Com-binatorial Structures and their Applications ed R Guy et al (Gordonand Breach New York) 403-406 [43]Stanley RP (1973) Acyclic orientations of graphs Discrete Math 5171-178 [70]

200 References

Temperley HNV (1964) On the mutual cancellation of cluster inte-grals in Mayers fugacity series Proc Phys Soc 83 3-16 [39]Thistlethwaite MB (1987) A spanning tree expansion of the Jonespolynomial Topology 26 297-309 [105]Thomassen C (1990) Resistances and currents in infinite electricalnetworks JCT(B) 49 87-102 [36]Tutte WT (1947a) A family of cubical graphs Proc CambridgePhilos Soc 45 459-474 [130141143]Tutte WT (1947b) A ring in graph theory Proc Cambridge PhilosSoc 43 26-40 [79]Tutte WT (1954) A contribution to the theory of chromatic polyno-mials Canad J Math 6 80-91 [30102110]Tutte WT (1956) A class of Abelian groups Canad J Math 813-28 [30]Tutte WT (1966) Connectivity in Graphs (Toronto University Press)

[120132 189]TutteWT(1967) On dichromatic polynomials JCT2 301-320

[81869096]Tutte WT (1970) On chromatic polynomials and the golden ratioJCT 9 289-296 [71]Tutte WT (1974) Codichromatic graphs JCT(B) 16 168-174 [79]Tutte WT (1979) All the kings horses - a guide to reconstructionGraph Theory and Related Topics ed JA Bondy and USR Murty(Academic Press) 15-33 [57]Tutte WT (1984) Graph Theory (Addison-Wesley) [72]Vijayan KS (1972) Association schemes and Moore graphs NoticesAmer Math Soc 19 A-685 [182]Weiss A (1984) Girths of bipartite sextet graphs Combinatorica 4241-245 [190]Weiss R (1974) Uber s-regulare Graphen JCT(B) 16 229-233 [141]Weiss R (1983) The non-existence of 8-transitive graphs Combina-torica 1 309-311 [137]Weiss R (1985) On distance-transitive graphs Bull London MathSoc 17 253-256 [163]Whitney H (1932a) A logical expansion in mathematics Bull AmerMath Soc 38 572-579 [7677]Whitney H (1932b) The coloring of graphs Ann Math 33 688-718 [698186]Whitney H (1932c) Congruent graphs and the connectivity of graphsAmer J Math 54 150-168 [120]

References 201

Wielandt H (1964) Finite Permutation Groups (Academic Press)[161172]

Wilf HS (1967) The eigenvalues of a graph and its chromatic numberJ London Math Soc 42 330-332 [55]Wilf HS (1986) Spectral bounds for the clique and independencenumbers of graphs JCT(B) 40 113-117 [59]Woodall DR (1992) A zero-free interval for chromatic polynomialsDiscrete Math 101 333-341 [72]Yetter DN (1990) On graph invariants given by linear recurrencerelations JCT(B) 48 6-18 [72]Yuan Hong (1988) A bound on the spectral radius of graphs Lin AlgAppl 108 135-139 [13]

Index

acyclic orientation 70adjacent 7adjacency algebra 9adjacency matrix 7almost-complete 43alternating knot 105angles 51antipodal 177antipodal r-fold covering 178augmentation 29automorphic 178automorphism 115automorphism group 115

bicentroid 119bigrading 97bipartite 11biplane 189block 81block system 173broken cycle 77Brookss theorem 55buckminsterfullerene 127

cage 181 188 189Cayley graph 123centroid 119characteristic polynomial 8chromatically unique 69chromatic invariant 107chromatic number 52chromatic polynomial 63chromatic root 71circulant graph 16 126circulant matrix 16closed walk 12coboundary mapping 28cocktail-party graph 17 68colour-class 52colour-partition 52compatible 150complete bipartite graph 21

complete graph 8complete matching 50complete multipartite graph 41conductance 34cone 66confluence 72conforms 30conjugate Bell polynomials 72connected 10contracting 64Conways presentations 145co-rank 25 97coset graph 128cospectral graphs 12 49cover 50covering graph 149cube 43 69 140 157 161 169cubic graph 138current 34cut 26cut-orientation 26cut-subspace 26cut-vertex 67cycle 25cycle graph 17 65cycle-orientation 25cycle-subspace 26

degree 4deletion-contraction 65 72density 94derived graph 178Desargues graph 148 153diameter 10dihedral group 126distance 10distance matrices 13 159distance-regular 13 159distance-transitive 118 155dodecahedron 69 178double pyramid 68

Index 203

double-transitivity 118dual 2943

edge 3edge space 23edge-transitive 115 118 120effective resistance 36eigenvalue 8electrical network 34elementary 44ends 4equipartition 58even subgraph 110excess 28 189expansion 147external activity 99externally active 99

feasible array 168flow 29flow polynomial 110forest 47Fosters census 147friendship theorem 171

generalized d-gon 187generalized line graph 21generalized polygon graph 181general graph 3girth 28 76 131 180graph 4graphical regular representation

124 128graph types 87

Hamiltonian cycle 50Hamming graph 169Heawood graph 148 154 163Hoffman-Singleton graph 189homeomorphic 79 108homogeneous 120homological covering 154Hopf algebra 88hyperoctahedral graph 17

icosahedral group 127icosahedron 69 178imprimitive 177

incidence mapping 24 29incidence matrix 24independent 98indicator function 74induced subgraph 4interaction model 80internal activity 99internally active 99intersection array 157 159intersection matrix 165intersection numbers 156Ising model 80isoperimetric number 28 58isthmus 30

join 66Jones polynomial 105

K-chain 149Kellys lemma 50Kirchhoffs laws 34Kocays lemma 50Krein parameters 170

labelled tree 104ladder 69 126Laplacian matrix 27Laplacian spectrum 29 40line graph 17 120logarithmic transform 82loop 3

medial graph 104minimal support 29Mobius ladder 20 42 69 110modified rank polynomial 101modular flow 30Moore graph 181Motzkin-Straus formula 59

negative end 24nowhere-zero 30

octahedron 43odd graphs 20 58 137 161 170orbit 115orientation 24

Paley graph 129

204 Index

Pappus graph 148 154partial geometry 162partition function 80path graph 11perfect code 22 171permutation character 172permutation matrix 116Petersen graph 20 95 103 133planar 29positive end 24potential 36Potts model 80power 36primitive 30 173principal minors 8projective plane 163proper 90pyramid 68

quasi-separable 67quasi-separation 67

rank 25rank matrix 73rank polynomial 73Rayleigh quotient 54Rayleighs monotonicity law 37reconstructible 50 91reconstruction conjecture 50recursive family 70 103regular graph 14regular action 122resonant model 80rewriting rules 72root systems 22r-ply transitive 162

semi-direct product 150separable 67separation 67series-parallel 109sextet graph 145Shannon capacity 51sides 149simple eigenvalues 116 125spanning elementary subgraph 44spanning tree 31spectral decomposition 13

spectrum 8sporadic groups 172square lattice 96stabilizer 122stabilizer sequence 133 137 147standard bases 24star graph 49star types 87strict graph 4strongly regular graph 16 20 159

171subdividing 79subgraph 4successor 132support 29suspension 66symmetric 118 126symmetric cycle 137symmetric design 163symmetric group 118 148

t-arc 130tetrahedral group 127thermodynamic limit 94theta graph 86Thomsons principle 36topological invariant 79totally unimodular 34tree 47 49 65 119tree-number 38triangle graph 19 169tridiagonal 165t-transitive 131Turans Theorem 59Tutte polynomial 97 100

umbral chromatic polynomial 72unimodal conjecture 108

vertex 3vertex-colouring 52vertex space 23vertex-stabilizer 122 127vertex-transitive 115 120 125V-function 79voltage 34

walk 9

Index 205

walk-generating function 13 weakly homogeneous 120walk-generating matrix 12 wheel 68


Second Edition

NORMAN BIGGS

London School of Economics

CAMBRIDGEUNIVERSITY PRESS











ISBN 978-0-521-20335-7 Hardback



Contents

Preface vii

1 Introduction 1







vi Contents



Preface







About the book





These are



Outline of the book











PART ONE













A =


3 - 1

-011

1

1011

1101

1-110

Spec K4 = bdquo


X(F A) = An + dA1 + c2A~2 + C3A-3 + + cn










010

100

000

5

011

100

100

011

101

110






such that Ut_i and

Vi = U0UiUi = Vj










x = wowiWd = y








Additional Results







I B








r = l


trG(z) =










a=0


n-l

a=0



It follows that







ZJ X$ ~-mdash KXj





yj = XyiltVyiltkyj















3=2







Spec _(n- - 1 Kn~ 1 n-l)-

1 J







s - 2 0 - 2





f l Xvi 0 othe




Proof (1) We have









X(L(T) A) = (A + 2)m-nX(T X + 2-k)









= det((A + 2)Im - XX)

= (A + 2)m-det((A + 2)In - XX4)

= (A + 2)m-det((A + 2- jfe)In - A)

D


k X As_Spec r =




2 t - 4 i - 4 - 2


Additional results




3 1 - 2 s

Spec O3 = 1 g



A2 + (c - o)A + (c - fc)I = cJ













13










d lt 2 2k V log2n



ln(n-l)d lt

Cycles and cuts






y = [yiy2---ynt





defined by









D(1) 0 00 Dlt2gt 0

0 0 D^c


Cycles and cuts 25






















Cycles and cuts 27







Q = DD = A - A




Additional Results








fo(x) = 1 fx(x)=x f2(x) = x2-k

fq(x) - xfq-xx) ~(k- l) -2(i) (q gt 3)




9ix) = 0(a) + fix) + + fi(x)

Cycles and cuts 29















on S((fgt)







mour 1981)
























r bull




CT 1c=

bulllm-n+1




X-ir


JX = (Uy UN) bull








Dw = 0 Cz = 0


[ wj-1w = z =






Additional Results













z = w + n Dw = 0 Cz = 0















R(xyV) gt R(xyT)


6

The tree-number










The tree-number 39



adj Q = K(F)J


det(D[)det(Dpound)





6(D = n-2det (J + Q)


(nl - J)(J + Q) = nJ + nQ - J2 - JQ = nQ







det (J + Q)J = n2Ki





mi ms-i

then


r=l






x(L(r)2k-2) = (2k)m-nx(Tk)


The tree-number 41







K(T) =n-2aTcn)






bull



aKa- n) = (-l)ax(Ka a - 1 - fj) = - a)01


K(Kaua2a) = n-2(n)(n - a^11 (n)(n - CL)0-1

= n ~2 (n - ai)ai-1 (n - a




Additional Results


lt --t I N nmdash1

1 I nkn n mdash 1



n - l





1 2h~1


nMh) = ~[(2 + v3) + (2 - v3)h] + h


The tree-number 43









(n - M)ltr(FcM) = (-1)














r(A) =n~(No + Ne) == Ne (mod 2)




det A(K4) = 3(-l)22deg + 3(-l)321 = - 3


X(F A) - Xn + cxA1 + c2A-2 + + cn






C4 = na - 2rib






We shall write


















XY













Additional Results














(rA) pound ( r A )








aij=Eiej (lltijltn)





8












X(F z) = zn + c2zn~2 + CiZ1- + = z6pz2)



A - f deg JA ~ [J 0



























uT) lt 1 + Amax(r)



bull







P QlQ R j










t














Additional Results




ltrgt s



Mi = min i(Qx)ux=O


vweE v





Mi lt SX ( 2





= 1


2(2r23) k



problem (QP)



1 gtw(r) - n2



PART TWO

Colouring problems

9






bull




Hence

C(Kn u) = uu - l)(w - 2) (u - n + 1)
























are given by

mi(T) =





D


+ 3U(2) + U(i)) O (u(3) + 3U(2) + U

6U(5) + llU(4) + 6W(3) + W(2)

M5 + 36w4 - 75u3 + 78u2 - 31u





C(cT u) = uC(F u - 1)

C(sT u) = uu - 1)C(F u - 2) + uC(T u - 1)



j(w - l)(j) = uC(Tu - 1)








C(Fu) =



C(Ta)






Additional Results


C(Wn u) = u(u - 2)71-1 -(- ( - l ) - 1 ^ - 2)






C(Lh u) = (u2 -3u + 3)h + (u- 1)(3 - u)h + (1 - u)h) + u2 -

C(Mh u) = (u2 -3u + 3)h + (u - 1)(3 - u)h - (1 - u)h - 1






u8 -24u7+260u6 - 1670u5+6999w4 - 19698u3+36408u2 -40240u+20170




-u(u - l)(w - 2) ] T Ci(l - u)


1 10 56 230 759 2112 5104 10912 20880 35972

55768 77152 93538 96396 80572 50808 21302 4412


k

Pk = ^ m r f c ( r ) r=0


2fc=0 V r = 0 fc=0 V




fe








mdash n~5

C(F (3 bull










Given a graph F let




10

Subgraph expansions



R(Txy) = Er




L 81 78 36 9 1J






QET- 01 as follows

_ f 1






C(Tu)=SCET




c(rlaquo)= X) E




|VT VS + c = n - VS + c = n- rS)

D





j




M = C(Tu) = unR(T -u~ -1)



bx = - 9 62 = 36 h = -84 + 9 = -75 b4 = 117 - 45 + 6 = 78

65 = -81 + 78 - 36 + 9 - 1 = -31


3t3 u)=u6 - 9M5 + 36M4 - 75u3 + 78u2


(1) (-l)lt6-=(7) for i = 0lg-2

(2) ( - l ) raquo - V i = ( f l - i ) - -



















Additional Results
















pound a - 1)SCEV U





11








X(T) - ]T YS)SCET






Y(S)r = Y(S1)r1Y(S2)r2-



S1CE1 S2CE2



ICJ








X(S) = X(S1)X(S2)


SCET



52 (-i)|S2|iog x(s2



XT)= n YS)SCEF




We shall prove that

iogx(r)=SCET







bull





bull
















Additional Results



1 - (1 - u)r-n - (1 - u)s~n - (1 - it)- + (2 - u)(l - u)~n

(1 - (1 - u)r-n)(l - (1 - u)s~n)(l - (1 - uf-n)





7 1 2

2 +











Explicitly we have





12





Q(Cnu) = q(Cnu) =



















Q(F u) = 1 + (30u~n+l + l U - n +



Q(K4u)


and so

C(K4 u) = u4- 6u3 + llu2 - au + b


C(K4 u) = u4 - 6w3 + llu2 - 6u = u(u - l)(u - 2)u - 3)







A Aa A A5



6 l j


A2 A3 A4

C u ( u - l ) u(u - l)(u2 - 3w + 3)

Q (u - l)u u(u - 2)(u - I)2 u(u2 -3u + 3)(u - I)3


= u(u - l)(u - 2)(w2 - 3w + 3)





(I)6

= u6 - 9u5 + 34u4 - 67u3 + 67w2






) = P(u)Q(Tu)



P(u) =C((V1DV2u)














Additional Results



where ) = (-ir-1-(T1)


r(K2u) = 1u

r(K3u)= -lu-lf

r(K4 u) = -(2u - 3)u(u - 2)2

r(C4u)= l (w- l ) 3 -


r(K5 u) = -(6u4 - 48w3 + 140u2 - 176u + 81)(u - l)4(u - 3)4

r(W5 u) = (3u2 - 9u + 7)u(u - 2f(u2 - 3u + 3)

r(W-u) = (2u2 - 6u + 5)u(u - 2)(M2 - 3u + 3)2

r(K23 u) = (u3 - 6u2 + llu - 7)u(u2 - Zu + 3)3

r(C5u)= -l(u-l)



5681012

15

1024612

112410


14 19 3 130 15 10 1








2 + v ^ - 4 u + 8)lt u


+ x7 + 3xs + 4x9 + 3x10 Ux 12

- 91a15 - 261a16 - 290x17

+ 24a13

254x18

8z14

13













r(X n(X)) = r(X) s(X M(X)) = s(X) - 1








AXA = X = xix2)xt



gtlUxi x a_i









X+=XUXeuro X~=XXL









A - ^ B

VC - ^ V












ni = W 6 C | r(W) = r0 - t |W~| = j |











cients

k x 7 i










Additional Results





r 0 36 84 75 35 9 136 168 171 65 10120 240 105 15180 170 30170 70114 1256216

L l










r(xya) =x-1



A







TYxy)=TTyx)


fcgt0








14




C(T u) = (-l)-1^ ]T laquo(1 - )t=i



C(Tu) =unR(F-u-1-l)






n-2






bull








0(1^) = 0(T2)



C(rl) = -C(r ( e ) l )













^ n-l-A ^ (n~l-l



Additional Results









0(Kn) = (n - 2) (n gt 2)

0(Lh) = 2fe - ft - 1 (ft gt 3)

0(Mh) = 2h - ft (ft gt 2)




C(Tu)=uC(ru)









C ( F (u + y - 1 ) 2 ) = ] pound ( V c




log TB(n)log TAn)


PART THREE


15













r D














duv) = dau)av))






Additional Results























16





G Gv = GGV = |VT|








l (12)

(123)

(132) (23)




hkGET=gth~lkefl

= (gh)~lgk euro Q




a


































Additional Results
























17

Symmetric graphs











Pg-t=O0-

[a]


























Thus we have




















Additional Results



Jjy LtyLiU

jLjyyj J-JW


GV lt k(k - 1)LVW


G lt 2nGv lt 2nn (n - 1) = 2 (n)2











18











111 = 2 ( 0 lt t lt t - l )

|Ft| = 3x2-1



9 = 9i ^o = 9T192 xi=g~lxag (i = 1 2 )


bull bull [a]












x0 = (34) Xl = (12) x2 = (35) x3 = (14)



5 (i

8 7


51 = (1234)(5678) g2 = (123785)(46)

xo - (36)(45) X = (16)(47) x2 =













as claimed D


[XiXj] = (XiXj)2

















1 [xoxx][xo OM

= [XoXx]







| ( laquo - l ) lt A lt | ( t + 2)




2)2

= (x5x0x5)2 = x5xlx5 = 1








|Aut(fi)| = |AutS6| = 1440 = 30 x 3 x 24




x3 = |(o6)(c)(de)| x4 = |(c)|


F5 = 54xZ2 F4 = D8x Z2 F3 = (Z2)3

F2 = (Z2)2 F=Z2


Additional Results




^(a - d)(b - c)






n = ^rp(p2 - 1) when p = 79 (mod 16)

n = ^p2ipA - 1) when p = 351113 (mod 16)









a2 = 1 (era)2 = 1 (ab)2 = 1 (a^b)2 = 1 abaa2 = b2


a2 = 1 aaf = 1 (crb)2 = 1 (a1))2 = 1 ( a~V) 2 = 1

(a-363)2 = 1 a4b4a4 = ba a4baab = ba3b







Ft = (a-ibii= 12t)





t Fl F2 F3 F4 F5

1 Z 3

2 Z 2 S3

3 Z 2 (Z2)2 D12

4 Z2 (Z2)2 As2 (Z2)2

)5 Z2 (Z2)2










a = (12) (34675) (89X) b = (1246853)(79X) a = (12)(34)(56)(9JsT)


19









r f






KST

ST -1+ K




(K9)(KV) ~ (Kg(n)g(v))
















V - U0Ui U[ = V








bull



Additional Results






ltfgt(vw)=je







20











D





Shij = Shi(uv)











C i Cj

a0 ai a bo bi bj


1 2 3t(Q3) = 0 0 0 0

3 2 1








f 1 ) ( I ktKn) ~ 0 n - 2 i(Kkik) = 0 0 0


1 4

0 t-2 2 - 82f - 4 t - 3


















-Imdash 1


kh bd-i 1 c 2 ca









(A - AI)(Ao + Ai + + Ad) = 0





d

j=o


rs mdash Shi(vrvs)





Additional Results




21 - 121 - 221 - 2 1 + 11 + 11 1122 1 - 11 - 1

2121 - 121 - 1 I + 11 + 1 1122 I - 11 - 1









s(t + 1) (s - a + l)t 1 a(t + 1)




^CA I K A 1 A fe j




21















TO 1k d i C2

h a2 bull

bull bull cd

bull adA








vo() = 1










BK = (BK)( - KB



D





^) ioltiltd)


3=0











n3




0 1 02r r 40 r - 1 2r - 4




v0 =

12r

gt(r-l) Vi =

1r - 21 - r

v2 = 1 - 21

m(Ai) n(ui

m(A2) =

1 + (r - 2)22r + (1 - r)2rr - 1)

1 i r ( r _ 1 N = 2~(r ~ 1)(r(u2v2) 1-



_ _ 16 4 -2s



Additional Results



d(q-l)(d-l)(q-l)(q-l) 12 d




i) =m




n(0 lt i lt d)



=E-(r = 0



B mdash090

108

045


0810

1512




n lt -m(m + 3)













22

Imprimitivity

















Imprimitivity 175






















Imprimitivity 177

















Additional Results



i(I) = 521 125 i(D) = 32111 11123



Imprimitivity 179












(O6)H


+ (15)(23)(46)- (12)(36)(45)- (14)(25)(36)-gt (16)(23)(45)- (15)(26)(34)

(ac)i-

(laquo)-(ce) H

-gt (14)(26)(35)-raquo (16)(25)(34)^ (16)(24)(35)- (13)(25)(46)- (12)(35)(46)

(ad)v-(6c) H(bf)-(c)( e ) -

bull+ (13)(24)(56)-gt (12)(34)(56)- (13)(26)(45)- (15)(24)(36)- (14)(23)(56)


23




no(kg) = l + k + k(k-l) + + k(k- l)^g-3l


fcfc-lfc-lfc-lllll



k k ~ 1 k 1 K 1 1 1 1 1 kj







fcl and fcfc-lljfc






rok

10

J f c - 110- 1

10

fc-1c



() cud-i(X) + (k - c - )ud) = 0




(1 lt i lt d)








k 2gcosa (j = 12 d)







2





F = 2h-(h2 + l)cos2a G = (h2 - 1)sin2a



)



nk (4h-2



















m _ (fc + fc3)(4fc-4-A2)W (Jfe-A)(6-2 + 5A)


s5 + s4 + 6s3 - 2s2 + (9 - mi)s - 15 = 0







m = k = -(A3 - fc2 + fc)








lt2qx - ( 2





we have





R(X) = gm(X)nk = (4ft - X2)(k - X)(f + A)



(R(X) - 1)A2 + R(X)(f - k)X - (R(X)fk - Ah) = 0











Additional Results


















REFERENCES


192 References


References 193



194 References





References 195




196 References



References 197


198 References


References 199


200 References




References 201



Index






Index 203















Paley graph 129

204 Index










walk 9

Index 205












ISBN 978-0-521-20335-7 Hardback



Contents

Preface vii

1 Introduction 1







vi Contents



Preface







About the book





These are



Outline of the book











PART ONE













A =


3 - 1

-011

1

1011

1101

1-110

Spec K4 = bdquo


X(F A) = An + dA1 + c2A~2 + C3A-3 + + cn










010

100

000

5

011

100

100

011

101

110






such that Ut_i and

Vi = U0UiUi = Vj










x = wowiWd = y








Additional Results







I B








r = l


trG(z) =










a=0


n-l

a=0



It follows that







ZJ X$ ~-mdash KXj





yj = XyiltVyiltkyj















3=2







Spec _(n- - 1 Kn~ 1 n-l)-

1 J







s - 2 0 - 2





f l Xvi 0 othe




Proof (1) We have









X(L(T) A) = (A + 2)m-nX(T X + 2-k)









= det((A + 2)Im - XX)

= (A + 2)m-det((A + 2)In - XX4)

= (A + 2)m-det((A + 2- jfe)In - A)

D


k X As_Spec r =




2 t - 4 i - 4 - 2


Additional results




3 1 - 2 s

Spec O3 = 1 g



A2 + (c - o)A + (c - fc)I = cJ













13










d lt 2 2k V log2n



ln(n-l)d lt

Cycles and cuts






y = [yiy2---ynt





defined by









D(1) 0 00 Dlt2gt 0

0 0 D^c


Cycles and cuts 25






















Cycles and cuts 27







Q = DD = A - A




Additional Results








fo(x) = 1 fx(x)=x f2(x) = x2-k

fq(x) - xfq-xx) ~(k- l) -2(i) (q gt 3)




9ix) = 0(a) + fix) + + fi(x)

Cycles and cuts 29















on S((fgt)







mour 1981)
























r bull




CT 1c=

bulllm-n+1




X-ir


JX = (Uy UN) bull








Dw = 0 Cz = 0


[ wj-1w = z =






Additional Results













z = w + n Dw = 0 Cz = 0















R(xyV) gt R(xyT)


6

The tree-number










The tree-number 39



adj Q = K(F)J


det(D[)det(Dpound)





6(D = n-2det (J + Q)


(nl - J)(J + Q) = nJ + nQ - J2 - JQ = nQ







det (J + Q)J = n2Ki





mi ms-i

then


r=l






x(L(r)2k-2) = (2k)m-nx(Tk)


The tree-number 41







K(T) =n-2aTcn)






bull



aKa- n) = (-l)ax(Ka a - 1 - fj) = - a)01


K(Kaua2a) = n-2(n)(n - a^11 (n)(n - CL)0-1

= n ~2 (n - ai)ai-1 (n - a




Additional Results


lt --t I N nmdash1

1 I nkn n mdash 1



n - l





1 2h~1


nMh) = ~[(2 + v3) + (2 - v3)h] + h


The tree-number 43









(n - M)ltr(FcM) = (-1)














r(A) =n~(No + Ne) == Ne (mod 2)




det A(K4) = 3(-l)22deg + 3(-l)321 = - 3


X(F A) - Xn + cxA1 + c2A-2 + + cn






C4 = na - 2rib






We shall write


















XY













Additional Results














(rA) pound ( r A )








aij=Eiej (lltijltn)





8












X(F z) = zn + c2zn~2 + CiZ1- + = z6pz2)



A - f deg JA ~ [J 0



























uT) lt 1 + Amax(r)



bull







P QlQ R j










t














Additional Results




ltrgt s



Mi = min i(Qx)ux=O


vweE v





Mi lt SX ( 2





= 1


2(2r23) k



problem (QP)



1 gtw(r) - n2



PART TWO

Colouring problems

9






bull




Hence

C(Kn u) = uu - l)(w - 2) (u - n + 1)
























are given by

mi(T) =





D


+ 3U(2) + U(i)) O (u(3) + 3U(2) + U

6U(5) + llU(4) + 6W(3) + W(2)

M5 + 36w4 - 75u3 + 78u2 - 31u





C(cT u) = uC(F u - 1)

C(sT u) = uu - 1)C(F u - 2) + uC(T u - 1)



j(w - l)(j) = uC(Tu - 1)








C(Fu) =



C(Ta)






Additional Results


C(Wn u) = u(u - 2)71-1 -(- ( - l ) - 1 ^ - 2)






C(Lh u) = (u2 -3u + 3)h + (u- 1)(3 - u)h + (1 - u)h) + u2 -

C(Mh u) = (u2 -3u + 3)h + (u - 1)(3 - u)h - (1 - u)h - 1






u8 -24u7+260u6 - 1670u5+6999w4 - 19698u3+36408u2 -40240u+20170




-u(u - l)(w - 2) ] T Ci(l - u)


1 10 56 230 759 2112 5104 10912 20880 35972

55768 77152 93538 96396 80572 50808 21302 4412


k

Pk = ^ m r f c ( r ) r=0


2fc=0 V r = 0 fc=0 V




fe








mdash n~5

C(F (3 bull










Given a graph F let




10

Subgraph expansions



R(Txy) = Er




L 81 78 36 9 1J






QET- 01 as follows

_ f 1






C(Tu)=SCET




c(rlaquo)= X) E




|VT VS + c = n - VS + c = n- rS)

D





j




M = C(Tu) = unR(T -u~ -1)



bx = - 9 62 = 36 h = -84 + 9 = -75 b4 = 117 - 45 + 6 = 78

65 = -81 + 78 - 36 + 9 - 1 = -31


3t3 u)=u6 - 9M5 + 36M4 - 75u3 + 78u2


(1) (-l)lt6-=(7) for i = 0lg-2

(2) ( - l ) raquo - V i = ( f l - i ) - -



















Additional Results
















pound a - 1)SCEV U





11








X(T) - ]T YS)SCET






Y(S)r = Y(S1)r1Y(S2)r2-



S1CE1 S2CE2



ICJ








X(S) = X(S1)X(S2)


SCET



52 (-i)|S2|iog x(s2



XT)= n YS)SCEF




We shall prove that

iogx(r)=SCET







bull





bull
















Additional Results



1 - (1 - u)r-n - (1 - u)s~n - (1 - it)- + (2 - u)(l - u)~n

(1 - (1 - u)r-n)(l - (1 - u)s~n)(l - (1 - uf-n)





7 1 2

2 +











Explicitly we have





12





Q(Cnu) = q(Cnu) =



















Q(F u) = 1 + (30u~n+l + l U - n +



Q(K4u)


and so

C(K4 u) = u4- 6u3 + llu2 - au + b


C(K4 u) = u4 - 6w3 + llu2 - 6u = u(u - l)(u - 2)u - 3)







A Aa A A5



6 l j


A2 A3 A4

C u ( u - l ) u(u - l)(u2 - 3w + 3)

Q (u - l)u u(u - 2)(u - I)2 u(u2 -3u + 3)(u - I)3


= u(u - l)(u - 2)(w2 - 3w + 3)





(I)6

= u6 - 9u5 + 34u4 - 67u3 + 67w2






) = P(u)Q(Tu)



P(u) =C((V1DV2u)














Additional Results



where ) = (-ir-1-(T1)


r(K2u) = 1u

r(K3u)= -lu-lf

r(K4 u) = -(2u - 3)u(u - 2)2

r(C4u)= l (w- l ) 3 -


r(K5 u) = -(6u4 - 48w3 + 140u2 - 176u + 81)(u - l)4(u - 3)4

r(W5 u) = (3u2 - 9u + 7)u(u - 2f(u2 - 3u + 3)

r(W-u) = (2u2 - 6u + 5)u(u - 2)(M2 - 3u + 3)2

r(K23 u) = (u3 - 6u2 + llu - 7)u(u2 - Zu + 3)3

r(C5u)= -l(u-l)



5681012

15

1024612

112410


14 19 3 130 15 10 1








2 + v ^ - 4 u + 8)lt u


+ x7 + 3xs + 4x9 + 3x10 Ux 12

- 91a15 - 261a16 - 290x17

+ 24a13

254x18

8z14

13













r(X n(X)) = r(X) s(X M(X)) = s(X) - 1








AXA = X = xix2)xt



gtlUxi x a_i









X+=XUXeuro X~=XXL









A - ^ B

VC - ^ V












ni = W 6 C | r(W) = r0 - t |W~| = j |











cients

k x 7 i










Additional Results





r 0 36 84 75 35 9 136 168 171 65 10120 240 105 15180 170 30170 70114 1256216

L l










r(xya) =x-1



A







TYxy)=TTyx)


fcgt0








14




C(T u) = (-l)-1^ ]T laquo(1 - )t=i



C(Tu) =unR(F-u-1-l)






n-2






bull








0(1^) = 0(T2)



C(rl) = -C(r ( e ) l )













^ n-l-A ^ (n~l-l



Additional Results









0(Kn) = (n - 2) (n gt 2)

0(Lh) = 2fe - ft - 1 (ft gt 3)

0(Mh) = 2h - ft (ft gt 2)




C(Tu)=uC(ru)









C ( F (u + y - 1 ) 2 ) = ] pound ( V c




log TB(n)log TAn)


PART THREE


15













r D














duv) = dau)av))






Additional Results























16





G Gv = GGV = |VT|








l (12)

(123)

(132) (23)




hkGET=gth~lkefl

= (gh)~lgk euro Q




a


































Additional Results
























17

Symmetric graphs











Pg-t=O0-

[a]


























Thus we have




















Additional Results



Jjy LtyLiU

jLjyyj J-JW


GV lt k(k - 1)LVW


G lt 2nGv lt 2nn (n - 1) = 2 (n)2











18











111 = 2 ( 0 lt t lt t - l )

|Ft| = 3x2-1



9 = 9i ^o = 9T192 xi=g~lxag (i = 1 2 )


bull bull [a]












x0 = (34) Xl = (12) x2 = (35) x3 = (14)



5 (i

8 7


51 = (1234)(5678) g2 = (123785)(46)

xo - (36)(45) X = (16)(47) x2 =













as claimed D


[XiXj] = (XiXj)2

















1 [xoxx][xo OM

= [XoXx]







| ( laquo - l ) lt A lt | ( t + 2)




2)2

= (x5x0x5)2 = x5xlx5 = 1








|Aut(fi)| = |AutS6| = 1440 = 30 x 3 x 24




x3 = |(o6)(c)(de)| x4 = |(c)|


F5 = 54xZ2 F4 = D8x Z2 F3 = (Z2)3

F2 = (Z2)2 F=Z2


Additional Results




^(a - d)(b - c)






n = ^rp(p2 - 1) when p = 79 (mod 16)

n = ^p2ipA - 1) when p = 351113 (mod 16)









a2 = 1 (era)2 = 1 (ab)2 = 1 (a^b)2 = 1 abaa2 = b2


a2 = 1 aaf = 1 (crb)2 = 1 (a1))2 = 1 ( a~V) 2 = 1

(a-363)2 = 1 a4b4a4 = ba a4baab = ba3b







Ft = (a-ibii= 12t)





t Fl F2 F3 F4 F5

1 Z 3

2 Z 2 S3

3 Z 2 (Z2)2 D12

4 Z2 (Z2)2 As2 (Z2)2

)5 Z2 (Z2)2










a = (12) (34675) (89X) b = (1246853)(79X) a = (12)(34)(56)(9JsT)


19









r f






KST

ST -1+ K




(K9)(KV) ~ (Kg(n)g(v))
















V - U0Ui U[ = V








bull



Additional Results






ltfgt(vw)=je







20











D





Shij = Shi(uv)











C i Cj

a0 ai a bo bi bj


1 2 3t(Q3) = 0 0 0 0

3 2 1








f 1 ) ( I ktKn) ~ 0 n - 2 i(Kkik) = 0 0 0


1 4

0 t-2 2 - 82f - 4 t - 3


















-Imdash 1


kh bd-i 1 c 2 ca









(A - AI)(Ao + Ai + + Ad) = 0





d

j=o


rs mdash Shi(vrvs)





Additional Results




21 - 121 - 221 - 2 1 + 11 + 11 1122 1 - 11 - 1

2121 - 121 - 1 I + 11 + 1 1122 I - 11 - 1









s(t + 1) (s - a + l)t 1 a(t + 1)




^CA I K A 1 A fe j




21















TO 1k d i C2

h a2 bull

bull bull cd

bull adA








vo() = 1










BK = (BK)( - KB



D





^) ioltiltd)


3=0











n3




0 1 02r r 40 r - 1 2r - 4




v0 =

12r

gt(r-l) Vi =

1r - 21 - r

v2 = 1 - 21

m(Ai) n(ui

m(A2) =

1 + (r - 2)22r + (1 - r)2rr - 1)

1 i r ( r _ 1 N = 2~(r ~ 1)(r(u2v2) 1-



_ _ 16 4 -2s



Additional Results



d(q-l)(d-l)(q-l)(q-l) 12 d




i) =m




n(0 lt i lt d)



=E-(r = 0



B mdash090

108

045


0810

1512




n lt -m(m + 3)













22

Imprimitivity

















Imprimitivity 175






















Imprimitivity 177

















Additional Results



i(I) = 521 125 i(D) = 32111 11123



Imprimitivity 179












(O6)H


+ (15)(23)(46)- (12)(36)(45)- (14)(25)(36)-gt (16)(23)(45)- (15)(26)(34)

(ac)i-

(laquo)-(ce) H

-gt (14)(26)(35)-raquo (16)(25)(34)^ (16)(24)(35)- (13)(25)(46)- (12)(35)(46)

(ad)v-(6c) H(bf)-(c)( e ) -

bull+ (13)(24)(56)-gt (12)(34)(56)- (13)(26)(45)- (15)(24)(36)- (14)(23)(56)


23




no(kg) = l + k + k(k-l) + + k(k- l)^g-3l


fcfc-lfc-lfc-lllll



k k ~ 1 k 1 K 1 1 1 1 1 kj







fcl and fcfc-lljfc






rok

10

J f c - 110- 1

10

fc-1c



() cud-i(X) + (k - c - )ud) = 0




(1 lt i lt d)








k 2gcosa (j = 12 d)







2





F = 2h-(h2 + l)cos2a G = (h2 - 1)sin2a



)



nk (4h-2



















m _ (fc + fc3)(4fc-4-A2)W (Jfe-A)(6-2 + 5A)


s5 + s4 + 6s3 - 2s2 + (9 - mi)s - 15 = 0







m = k = -(A3 - fc2 + fc)








lt2qx - ( 2





we have





R(X) = gm(X)nk = (4ft - X2)(k - X)(f + A)



(R(X) - 1)A2 + R(X)(f - k)X - (R(X)fk - Ah) = 0











Additional Results


















REFERENCES


192 References


References 193



194 References





References 195




196 References



References 197


198 References


References 199


200 References




References 201



Index






Index 203















Paley graph 129

204 Index










walk 9

Index 205


Contents

Preface vii

1 Introduction 1







vi Contents



Preface







About the book





These are



Outline of the book











PART ONE













A =


3 - 1

-011

1

1011

1101

1-110

Spec K4 = bdquo


X(F A) = An + dA1 + c2A~2 + C3A-3 + + cn










010

100

000

5

011

100

100

011

101

110






such that Ut_i and

Vi = U0UiUi = Vj










x = wowiWd = y








Additional Results







I B








r = l


trG(z) =










a=0


n-l

a=0



It follows that







ZJ X$ ~-mdash KXj





yj = XyiltVyiltkyj















3=2







Spec _(n- - 1 Kn~ 1 n-l)-

1 J







s - 2 0 - 2





f l Xvi 0 othe




Proof (1) We have









X(L(T) A) = (A + 2)m-nX(T X + 2-k)









= det((A + 2)Im - XX)

= (A + 2)m-det((A + 2)In - XX4)

= (A + 2)m-det((A + 2- jfe)In - A)

D


k X As_Spec r =




2 t - 4 i - 4 - 2


Additional results




3 1 - 2 s

Spec O3 = 1 g



A2 + (c - o)A + (c - fc)I = cJ













13










d lt 2 2k V log2n



ln(n-l)d lt

Cycles and cuts






y = [yiy2---ynt





defined by









D(1) 0 00 Dlt2gt 0

0 0 D^c


Cycles and cuts 25






















Cycles and cuts 27







Q = DD = A - A




Additional Results








fo(x) = 1 fx(x)=x f2(x) = x2-k

fq(x) - xfq-xx) ~(k- l) -2(i) (q gt 3)




9ix) = 0(a) + fix) + + fi(x)

Cycles and cuts 29















on S((fgt)







mour 1981)
























r bull




CT 1c=

bulllm-n+1




X-ir


JX = (Uy UN) bull








Dw = 0 Cz = 0


[ wj-1w = z =






Additional Results













z = w + n Dw = 0 Cz = 0















R(xyV) gt R(xyT)


6

The tree-number










The tree-number 39



adj Q = K(F)J


det(D[)det(Dpound)





6(D = n-2det (J + Q)


(nl - J)(J + Q) = nJ + nQ - J2 - JQ = nQ







det (J + Q)J = n2Ki





mi ms-i

then


r=l






x(L(r)2k-2) = (2k)m-nx(Tk)


The tree-number 41







K(T) =n-2aTcn)






bull



aKa- n) = (-l)ax(Ka a - 1 - fj) = - a)01


K(Kaua2a) = n-2(n)(n - a^11 (n)(n - CL)0-1

= n ~2 (n - ai)ai-1 (n - a




Additional Results


lt --t I N nmdash1

1 I nkn n mdash 1



n - l





1 2h~1


nMh) = ~[(2 + v3) + (2 - v3)h] + h


The tree-number 43









(n - M)ltr(FcM) = (-1)














r(A) =n~(No + Ne) == Ne (mod 2)




det A(K4) = 3(-l)22deg + 3(-l)321 = - 3


X(F A) - Xn + cxA1 + c2A-2 + + cn






C4 = na - 2rib






We shall write


















XY













Additional Results














(rA) pound ( r A )








aij=Eiej (lltijltn)





8












X(F z) = zn + c2zn~2 + CiZ1- + = z6pz2)



A - f deg JA ~ [J 0



























uT) lt 1 + Amax(r)



bull







P QlQ R j










t














Additional Results




ltrgt s



Mi = min i(Qx)ux=O


vweE v





Mi lt SX ( 2





= 1


2(2r23) k



problem (QP)



1 gtw(r) - n2



PART TWO

Colouring problems

9






bull




Hence

C(Kn u) = uu - l)(w - 2) (u - n + 1)
























are given by

mi(T) =





D


+ 3U(2) + U(i)) O (u(3) + 3U(2) + U

6U(5) + llU(4) + 6W(3) + W(2)

M5 + 36w4 - 75u3 + 78u2 - 31u





C(cT u) = uC(F u - 1)

C(sT u) = uu - 1)C(F u - 2) + uC(T u - 1)



j(w - l)(j) = uC(Tu - 1)








C(Fu) =



C(Ta)






Additional Results


C(Wn u) = u(u - 2)71-1 -(- ( - l ) - 1 ^ - 2)






C(Lh u) = (u2 -3u + 3)h + (u- 1)(3 - u)h + (1 - u)h) + u2 -

C(Mh u) = (u2 -3u + 3)h + (u - 1)(3 - u)h - (1 - u)h - 1






u8 -24u7+260u6 - 1670u5+6999w4 - 19698u3+36408u2 -40240u+20170




-u(u - l)(w - 2) ] T Ci(l - u)


1 10 56 230 759 2112 5104 10912 20880 35972

55768 77152 93538 96396 80572 50808 21302 4412


k

Pk = ^ m r f c ( r ) r=0


2fc=0 V r = 0 fc=0 V




fe








mdash n~5

C(F (3 bull










Given a graph F let




10

Subgraph expansions



R(Txy) = Er




L 81 78 36 9 1J






QET- 01 as follows

_ f 1






C(Tu)=SCET




c(rlaquo)= X) E




|VT VS + c = n - VS + c = n- rS)

D





j




M = C(Tu) = unR(T -u~ -1)



bx = - 9 62 = 36 h = -84 + 9 = -75 b4 = 117 - 45 + 6 = 78

65 = -81 + 78 - 36 + 9 - 1 = -31


3t3 u)=u6 - 9M5 + 36M4 - 75u3 + 78u2


(1) (-l)lt6-=(7) for i = 0lg-2

(2) ( - l ) raquo - V i = ( f l - i ) - -



















Additional Results
















pound a - 1)SCEV U





11








X(T) - ]T YS)SCET






Y(S)r = Y(S1)r1Y(S2)r2-



S1CE1 S2CE2



ICJ








X(S) = X(S1)X(S2)


SCET



52 (-i)|S2|iog x(s2



XT)= n YS)SCEF




We shall prove that

iogx(r)=SCET







bull





bull
















Additional Results



1 - (1 - u)r-n - (1 - u)s~n - (1 - it)- + (2 - u)(l - u)~n

(1 - (1 - u)r-n)(l - (1 - u)s~n)(l - (1 - uf-n)





7 1 2

2 +











Explicitly we have





12





Q(Cnu) = q(Cnu) =



















Q(F u) = 1 + (30u~n+l + l U - n +



Q(K4u)


and so

C(K4 u) = u4- 6u3 + llu2 - au + b


C(K4 u) = u4 - 6w3 + llu2 - 6u = u(u - l)(u - 2)u - 3)







A Aa A A5



6 l j


A2 A3 A4

C u ( u - l ) u(u - l)(u2 - 3w + 3)

Q (u - l)u u(u - 2)(u - I)2 u(u2 -3u + 3)(u - I)3


= u(u - l)(u - 2)(w2 - 3w + 3)





(I)6

= u6 - 9u5 + 34u4 - 67u3 + 67w2






) = P(u)Q(Tu)



P(u) =C((V1DV2u)














Additional Results



where ) = (-ir-1-(T1)


r(K2u) = 1u

r(K3u)= -lu-lf

r(K4 u) = -(2u - 3)u(u - 2)2

r(C4u)= l (w- l ) 3 -


r(K5 u) = -(6u4 - 48w3 + 140u2 - 176u + 81)(u - l)4(u - 3)4

r(W5 u) = (3u2 - 9u + 7)u(u - 2f(u2 - 3u + 3)

r(W-u) = (2u2 - 6u + 5)u(u - 2)(M2 - 3u + 3)2

r(K23 u) = (u3 - 6u2 + llu - 7)u(u2 - Zu + 3)3

r(C5u)= -l(u-l)



5681012

15

1024612

112410


14 19 3 130 15 10 1








2 + v ^ - 4 u + 8)lt u


+ x7 + 3xs + 4x9 + 3x10 Ux 12

- 91a15 - 261a16 - 290x17

+ 24a13

254x18

8z14

13













r(X n(X)) = r(X) s(X M(X)) = s(X) - 1








AXA = X = xix2)xt



gtlUxi x a_i









X+=XUXeuro X~=XXL









A - ^ B

VC - ^ V












ni = W 6 C | r(W) = r0 - t |W~| = j |











cients

k x 7 i










Additional Results





r 0 36 84 75 35 9 136 168 171 65 10120 240 105 15180 170 30170 70114 1256216

L l










r(xya) =x-1



A







TYxy)=TTyx)


fcgt0








14




C(T u) = (-l)-1^ ]T laquo(1 - )t=i



C(Tu) =unR(F-u-1-l)






n-2






bull








0(1^) = 0(T2)



C(rl) = -C(r ( e ) l )













^ n-l-A ^ (n~l-l



Additional Results









0(Kn) = (n - 2) (n gt 2)

0(Lh) = 2fe - ft - 1 (ft gt 3)

0(Mh) = 2h - ft (ft gt 2)




C(Tu)=uC(ru)









C ( F (u + y - 1 ) 2 ) = ] pound ( V c




log TB(n)log TAn)


PART THREE


15













r D














duv) = dau)av))






Additional Results























16





G Gv = GGV = |VT|








l (12)

(123)

(132) (23)




hkGET=gth~lkefl

= (gh)~lgk euro Q




a


































Additional Results
























17

Symmetric graphs











Pg-t=O0-

[a]


























Thus we have




















Additional Results



Jjy LtyLiU

jLjyyj J-JW


GV lt k(k - 1)LVW


G lt 2nGv lt 2nn (n - 1) = 2 (n)2











18











111 = 2 ( 0 lt t lt t - l )

|Ft| = 3x2-1



9 = 9i ^o = 9T192 xi=g~lxag (i = 1 2 )


bull bull [a]












x0 = (34) Xl = (12) x2 = (35) x3 = (14)



5 (i

8 7


51 = (1234)(5678) g2 = (123785)(46)

xo - (36)(45) X = (16)(47) x2 =













as claimed D


[XiXj] = (XiXj)2

















1 [xoxx][xo OM

= [XoXx]







| ( laquo - l ) lt A lt | ( t + 2)




2)2

= (x5x0x5)2 = x5xlx5 = 1








|Aut(fi)| = |AutS6| = 1440 = 30 x 3 x 24




x3 = |(o6)(c)(de)| x4 = |(c)|


F5 = 54xZ2 F4 = D8x Z2 F3 = (Z2)3

F2 = (Z2)2 F=Z2


Additional Results




^(a - d)(b - c)






n = ^rp(p2 - 1) when p = 79 (mod 16)

n = ^p2ipA - 1) when p = 351113 (mod 16)









a2 = 1 (era)2 = 1 (ab)2 = 1 (a^b)2 = 1 abaa2 = b2


a2 = 1 aaf = 1 (crb)2 = 1 (a1))2 = 1 ( a~V) 2 = 1

(a-363)2 = 1 a4b4a4 = ba a4baab = ba3b







Ft = (a-ibii= 12t)





t Fl F2 F3 F4 F5

1 Z 3

2 Z 2 S3

3 Z 2 (Z2)2 D12

4 Z2 (Z2)2 As2 (Z2)2

)5 Z2 (Z2)2










a = (12) (34675) (89X) b = (1246853)(79X) a = (12)(34)(56)(9JsT)


19









r f






KST

ST -1+ K




(K9)(KV) ~ (Kg(n)g(v))
















V - U0Ui U[ = V








bull



Additional Results






ltfgt(vw)=je







20











D





Shij = Shi(uv)











C i Cj

a0 ai a bo bi bj


1 2 3t(Q3) = 0 0 0 0

3 2 1








f 1 ) ( I ktKn) ~ 0 n - 2 i(Kkik) = 0 0 0


1 4

0 t-2 2 - 82f - 4 t - 3


















-Imdash 1


kh bd-i 1 c 2 ca









(A - AI)(Ao + Ai + + Ad) = 0





d

j=o


rs mdash Shi(vrvs)





Additional Results




21 - 121 - 221 - 2 1 + 11 + 11 1122 1 - 11 - 1

2121 - 121 - 1 I + 11 + 1 1122 I - 11 - 1









s(t + 1) (s - a + l)t 1 a(t + 1)




^CA I K A 1 A fe j




21















TO 1k d i C2

h a2 bull

bull bull cd

bull adA








vo() = 1










BK = (BK)( - KB



D





^) ioltiltd)


3=0











n3




0 1 02r r 40 r - 1 2r - 4




v0 =

12r

gt(r-l) Vi =

1r - 21 - r

v2 = 1 - 21

m(Ai) n(ui

m(A2) =

1 + (r - 2)22r + (1 - r)2rr - 1)

1 i r ( r _ 1 N = 2~(r ~ 1)(r(u2v2) 1-



_ _ 16 4 -2s



Additional Results



d(q-l)(d-l)(q-l)(q-l) 12 d




i) =m




n(0 lt i lt d)



=E-(r = 0



B mdash090

108

045


0810

1512




n lt -m(m + 3)













22

Imprimitivity

















Imprimitivity 175






















Imprimitivity 177

















Additional Results



i(I) = 521 125 i(D) = 32111 11123



Imprimitivity 179












(O6)H


+ (15)(23)(46)- (12)(36)(45)- (14)(25)(36)-gt (16)(23)(45)- (15)(26)(34)

(ac)i-

(laquo)-(ce) H

-gt (14)(26)(35)-raquo (16)(25)(34)^ (16)(24)(35)- (13)(25)(46)- (12)(35)(46)

(ad)v-(6c) H(bf)-(c)( e ) -

bull+ (13)(24)(56)-gt (12)(34)(56)- (13)(26)(45)- (15)(24)(36)- (14)(23)(56)


23




no(kg) = l + k + k(k-l) + + k(k- l)^g-3l


fcfc-lfc-lfc-lllll



k k ~ 1 k 1 K 1 1 1 1 1 kj







fcl and fcfc-lljfc






rok

10

J f c - 110- 1

10

fc-1c



() cud-i(X) + (k - c - )ud) = 0




(1 lt i lt d)








k 2gcosa (j = 12 d)







2





F = 2h-(h2 + l)cos2a G = (h2 - 1)sin2a



)



nk (4h-2



















m _ (fc + fc3)(4fc-4-A2)W (Jfe-A)(6-2 + 5A)


s5 + s4 + 6s3 - 2s2 + (9 - mi)s - 15 = 0







m = k = -(A3 - fc2 + fc)








lt2qx - ( 2





we have





R(X) = gm(X)nk = (4ft - X2)(k - X)(f + A)



(R(X) - 1)A2 + R(X)(f - k)X - (R(X)fk - Ah) = 0











Additional Results


















REFERENCES


192 References


References 193



194 References





References 195




196 References



References 197


198 References


References 199


200 References




References 201



Index






Index 203















Paley graph 129

204 Index










walk 9

Index 205


vi Contents



Preface







About the book





These are



Outline of the book











PART ONE













A =


3 - 1

-011

1

1011

1101

1-110

Spec K4 = bdquo


X(F A) = An + dA1 + c2A~2 + C3A-3 + + cn










010

100

000

5

011

100

100

011

101

110






such that Ut_i and

Vi = U0UiUi = Vj










x = wowiWd = y








Additional Results







I B








r = l


trG(z) =










a=0


n-l

a=0



It follows that







ZJ X$ ~-mdash KXj





yj = XyiltVyiltkyj















3=2







Spec _(n- - 1 Kn~ 1 n-l)-

1 J







s - 2 0 - 2





f l Xvi 0 othe




Proof (1) We have









X(L(T) A) = (A + 2)m-nX(T X + 2-k)









= det((A + 2)Im - XX)

= (A + 2)m-det((A + 2)In - XX4)

= (A + 2)m-det((A + 2- jfe)In - A)

D


k X As_Spec r =




2 t - 4 i - 4 - 2


Additional results




3 1 - 2 s

Spec O3 = 1 g



A2 + (c - o)A + (c - fc)I = cJ













13










d lt 2 2k V log2n



ln(n-l)d lt

Cycles and cuts






y = [yiy2---ynt





defined by









D(1) 0 00 Dlt2gt 0

0 0 D^c


Cycles and cuts 25






















Cycles and cuts 27







Q = DD = A - A




Additional Results








fo(x) = 1 fx(x)=x f2(x) = x2-k

fq(x) - xfq-xx) ~(k- l) -2(i) (q gt 3)




9ix) = 0(a) + fix) + + fi(x)

Cycles and cuts 29















on S((fgt)







mour 1981)
























r bull




CT 1c=

bulllm-n+1




X-ir


JX = (Uy UN) bull








Dw = 0 Cz = 0


[ wj-1w = z =






Additional Results













z = w + n Dw = 0 Cz = 0















R(xyV) gt R(xyT)


6

The tree-number










The tree-number 39



adj Q = K(F)J


det(D[)det(Dpound)





6(D = n-2det (J + Q)


(nl - J)(J + Q) = nJ + nQ - J2 - JQ = nQ







det (J + Q)J = n2Ki





mi ms-i

then


r=l






x(L(r)2k-2) = (2k)m-nx(Tk)


The tree-number 41







K(T) =n-2aTcn)






bull



aKa- n) = (-l)ax(Ka a - 1 - fj) = - a)01


K(Kaua2a) = n-2(n)(n - a^11 (n)(n - CL)0-1

= n ~2 (n - ai)ai-1 (n - a




Additional Results


lt --t I N nmdash1

1 I nkn n mdash 1



n - l





1 2h~1


nMh) = ~[(2 + v3) + (2 - v3)h] + h


The tree-number 43









(n - M)ltr(FcM) = (-1)














r(A) =n~(No + Ne) == Ne (mod 2)




det A(K4) = 3(-l)22deg + 3(-l)321 = - 3


X(F A) - Xn + cxA1 + c2A-2 + + cn






C4 = na - 2rib






We shall write


















XY













Additional Results














(rA) pound ( r A )








aij=Eiej (lltijltn)





8












X(F z) = zn + c2zn~2 + CiZ1- + = z6pz2)



A - f deg JA ~ [J 0



























uT) lt 1 + Amax(r)



bull







P QlQ R j










t














Additional Results




ltrgt s



Mi = min i(Qx)ux=O


vweE v





Mi lt SX ( 2





= 1


2(2r23) k



problem (QP)



1 gtw(r) - n2



PART TWO

Colouring problems

9






bull




Hence

C(Kn u) = uu - l)(w - 2) (u - n + 1)
























are given by

mi(T) =





D


+ 3U(2) + U(i)) O (u(3) + 3U(2) + U

6U(5) + llU(4) + 6W(3) + W(2)

M5 + 36w4 - 75u3 + 78u2 - 31u





C(cT u) = uC(F u - 1)

C(sT u) = uu - 1)C(F u - 2) + uC(T u - 1)



j(w - l)(j) = uC(Tu - 1)








C(Fu) =



C(Ta)






Additional Results


C(Wn u) = u(u - 2)71-1 -(- ( - l ) - 1 ^ - 2)






C(Lh u) = (u2 -3u + 3)h + (u- 1)(3 - u)h + (1 - u)h) + u2 -

C(Mh u) = (u2 -3u + 3)h + (u - 1)(3 - u)h - (1 - u)h - 1






u8 -24u7+260u6 - 1670u5+6999w4 - 19698u3+36408u2 -40240u+20170




-u(u - l)(w - 2) ] T Ci(l - u)


1 10 56 230 759 2112 5104 10912 20880 35972

55768 77152 93538 96396 80572 50808 21302 4412


k

Pk = ^ m r f c ( r ) r=0


2fc=0 V r = 0 fc=0 V




fe








mdash n~5

C(F (3 bull










Given a graph F let




10

Subgraph expansions



R(Txy) = Er




L 81 78 36 9 1J






QET- 01 as follows

_ f 1






C(Tu)=SCET




c(rlaquo)= X) E




|VT VS + c = n - VS + c = n- rS)

D





j




M = C(Tu) = unR(T -u~ -1)



bx = - 9 62 = 36 h = -84 + 9 = -75 b4 = 117 - 45 + 6 = 78

65 = -81 + 78 - 36 + 9 - 1 = -31


3t3 u)=u6 - 9M5 + 36M4 - 75u3 + 78u2


(1) (-l)lt6-=(7) for i = 0lg-2

(2) ( - l ) raquo - V i = ( f l - i ) - -



















Additional Results
















pound a - 1)SCEV U





11








X(T) - ]T YS)SCET






Y(S)r = Y(S1)r1Y(S2)r2-



S1CE1 S2CE2



ICJ








X(S) = X(S1)X(S2)


SCET



52 (-i)|S2|iog x(s2



XT)= n YS)SCEF




We shall prove that

iogx(r)=SCET







bull





bull
















Additional Results



1 - (1 - u)r-n - (1 - u)s~n - (1 - it)- + (2 - u)(l - u)~n

(1 - (1 - u)r-n)(l - (1 - u)s~n)(l - (1 - uf-n)





7 1 2

2 +











Explicitly we have





12





Q(Cnu) = q(Cnu) =



















Q(F u) = 1 + (30u~n+l + l U - n +



Q(K4u)


and so

C(K4 u) = u4- 6u3 + llu2 - au + b


C(K4 u) = u4 - 6w3 + llu2 - 6u = u(u - l)(u - 2)u - 3)







A Aa A A5



6 l j


A2 A3 A4

C u ( u - l ) u(u - l)(u2 - 3w + 3)

Q (u - l)u u(u - 2)(u - I)2 u(u2 -3u + 3)(u - I)3


= u(u - l)(u - 2)(w2 - 3w + 3)





(I)6

= u6 - 9u5 + 34u4 - 67u3 + 67w2






) = P(u)Q(Tu)



P(u) =C((V1DV2u)














Additional Results



where ) = (-ir-1-(T1)


r(K2u) = 1u

r(K3u)= -lu-lf

r(K4 u) = -(2u - 3)u(u - 2)2

r(C4u)= l (w- l ) 3 -


r(K5 u) = -(6u4 - 48w3 + 140u2 - 176u + 81)(u - l)4(u - 3)4

r(W5 u) = (3u2 - 9u + 7)u(u - 2f(u2 - 3u + 3)

r(W-u) = (2u2 - 6u + 5)u(u - 2)(M2 - 3u + 3)2

r(K23 u) = (u3 - 6u2 + llu - 7)u(u2 - Zu + 3)3

r(C5u)= -l(u-l)



5681012

15

1024612

112410


14 19 3 130 15 10 1








2 + v ^ - 4 u + 8)lt u


+ x7 + 3xs + 4x9 + 3x10 Ux 12

- 91a15 - 261a16 - 290x17

+ 24a13

254x18

8z14

13













r(X n(X)) = r(X) s(X M(X)) = s(X) - 1








AXA = X = xix2)xt



gtlUxi x a_i









X+=XUXeuro X~=XXL









A - ^ B

VC - ^ V












ni = W 6 C | r(W) = r0 - t |W~| = j |











cients

k x 7 i










Additional Results





r 0 36 84 75 35 9 136 168 171 65 10120 240 105 15180 170 30170 70114 1256216

L l










r(xya) =x-1



A







TYxy)=TTyx)


fcgt0








14




C(T u) = (-l)-1^ ]T laquo(1 - )t=i



C(Tu) =unR(F-u-1-l)






n-2






bull








0(1^) = 0(T2)



C(rl) = -C(r ( e ) l )













^ n-l-A ^ (n~l-l



Additional Results









0(Kn) = (n - 2) (n gt 2)

0(Lh) = 2fe - ft - 1 (ft gt 3)

0(Mh) = 2h - ft (ft gt 2)




C(Tu)=uC(ru)









C ( F (u + y - 1 ) 2 ) = ] pound ( V c




log TB(n)log TAn)


PART THREE


15













r D














duv) = dau)av))






Additional Results























16





G Gv = GGV = |VT|








l (12)

(123)

(132) (23)




hkGET=gth~lkefl

= (gh)~lgk euro Q




a


































Additional Results
























17

Symmetric graphs











Pg-t=O0-

[a]


























Thus we have




















Additional Results



Jjy LtyLiU

jLjyyj J-JW


GV lt k(k - 1)LVW


G lt 2nGv lt 2nn (n - 1) = 2 (n)2











18











111 = 2 ( 0 lt t lt t - l )

|Ft| = 3x2-1



9 = 9i ^o = 9T192 xi=g~lxag (i = 1 2 )


bull bull [a]












x0 = (34) Xl = (12) x2 = (35) x3 = (14)



5 (i

8 7


51 = (1234)(5678) g2 = (123785)(46)

xo - (36)(45) X = (16)(47) x2 =













as claimed D


[XiXj] = (XiXj)2

















1 [xoxx][xo OM

= [XoXx]







| ( laquo - l ) lt A lt | ( t + 2)




2)2

= (x5x0x5)2 = x5xlx5 = 1








|Aut(fi)| = |AutS6| = 1440 = 30 x 3 x 24




x3 = |(o6)(c)(de)| x4 = |(c)|


F5 = 54xZ2 F4 = D8x Z2 F3 = (Z2)3

F2 = (Z2)2 F=Z2


Additional Results




^(a - d)(b - c)






n = ^rp(p2 - 1) when p = 79 (mod 16)

n = ^p2ipA - 1) when p = 351113 (mod 16)









a2 = 1 (era)2 = 1 (ab)2 = 1 (a^b)2 = 1 abaa2 = b2


a2 = 1 aaf = 1 (crb)2 = 1 (a1))2 = 1 ( a~V) 2 = 1

(a-363)2 = 1 a4b4a4 = ba a4baab = ba3b







Ft = (a-ibii= 12t)





t Fl F2 F3 F4 F5

1 Z 3

2 Z 2 S3

3 Z 2 (Z2)2 D12

4 Z2 (Z2)2 As2 (Z2)2

)5 Z2 (Z2)2










a = (12) (34675) (89X) b = (1246853)(79X) a = (12)(34)(56)(9JsT)


19









r f






KST

ST -1+ K




(K9)(KV) ~ (Kg(n)g(v))
















V - U0Ui U[ = V








bull



Additional Results






ltfgt(vw)=je







20











D





Shij = Shi(uv)











C i Cj

a0 ai a bo bi bj


1 2 3t(Q3) = 0 0 0 0

3 2 1








f 1 ) ( I ktKn) ~ 0 n - 2 i(Kkik) = 0 0 0


1 4

0 t-2 2 - 82f - 4 t - 3


















-Imdash 1


kh bd-i 1 c 2 ca









(A - AI)(Ao + Ai + + Ad) = 0





d

j=o


rs mdash Shi(vrvs)





Additional Results




21 - 121 - 221 - 2 1 + 11 + 11 1122 1 - 11 - 1

2121 - 121 - 1 I + 11 + 1 1122 I - 11 - 1









s(t + 1) (s - a + l)t 1 a(t + 1)




^CA I K A 1 A fe j




21















TO 1k d i C2

h a2 bull

bull bull cd

bull adA








vo() = 1










BK = (BK)( - KB



D





^) ioltiltd)


3=0











n3




0 1 02r r 40 r - 1 2r - 4




v0 =

12r

gt(r-l) Vi =

1r - 21 - r

v2 = 1 - 21

m(Ai) n(ui

m(A2) =

1 + (r - 2)22r + (1 - r)2rr - 1)

1 i r ( r _ 1 N = 2~(r ~ 1)(r(u2v2) 1-



_ _ 16 4 -2s



Additional Results



d(q-l)(d-l)(q-l)(q-l) 12 d




i) =m




n(0 lt i lt d)



=E-(r = 0



B mdash090

108

045


0810

1512




n lt -m(m + 3)













22

Imprimitivity

















Imprimitivity 175






















Imprimitivity 177

















Additional Results



i(I) = 521 125 i(D) = 32111 11123



Imprimitivity 179












(O6)H


+ (15)(23)(46)- (12)(36)(45)- (14)(25)(36)-gt (16)(23)(45)- (15)(26)(34)

(ac)i-

(laquo)-(ce) H

-gt (14)(26)(35)-raquo (16)(25)(34)^ (16)(24)(35)- (13)(25)(46)- (12)(35)(46)

(ad)v-(6c) H(bf)-(c)( e ) -

bull+ (13)(24)(56)-gt (12)(34)(56)- (13)(26)(45)- (15)(24)(36)- (14)(23)(56)


23




no(kg) = l + k + k(k-l) + + k(k- l)^g-3l


fcfc-lfc-lfc-lllll



k k ~ 1 k 1 K 1 1 1 1 1 kj







fcl and fcfc-lljfc






rok

10

J f c - 110- 1

10

fc-1c



() cud-i(X) + (k - c - )ud) = 0




(1 lt i lt d)








k 2gcosa (j = 12 d)







2





F = 2h-(h2 + l)cos2a G = (h2 - 1)sin2a



)



nk (4h-2



















m _ (fc + fc3)(4fc-4-A2)W (Jfe-A)(6-2 + 5A)


s5 + s4 + 6s3 - 2s2 + (9 - mi)s - 15 = 0







m = k = -(A3 - fc2 + fc)








lt2qx - ( 2





we have





R(X) = gm(X)nk = (4ft - X2)(k - X)(f + A)



(R(X) - 1)A2 + R(X)(f - k)X - (R(X)fk - Ah) = 0











Additional Results


















REFERENCES


192 References


References 193



194 References





References 195




196 References



References 197


198 References


References 199


200 References




References 201



Index






Index 203















Paley graph 129

204 Index










walk 9

Index 205


Preface







About the book





These are



Outline of the book











PART ONE













A =


3 - 1

-011

1

1011

1101

1-110

Spec K4 = bdquo


X(F A) = An + dA1 + c2A~2 + C3A-3 + + cn










010

100

000

5

011

100

100

011

101

110






such that Ut_i and

Vi = U0UiUi = Vj










x = wowiWd = y








Additional Results







I B








r = l


trG(z) =










a=0


n-l

a=0



It follows that







ZJ X$ ~-mdash KXj





yj = XyiltVyiltkyj















3=2







Spec _(n- - 1 Kn~ 1 n-l)-

1 J







s - 2 0 - 2





f l Xvi 0 othe




Proof (1) We have









X(L(T) A) = (A + 2)m-nX(T X + 2-k)









= det((A + 2)Im - XX)

= (A + 2)m-det((A + 2)In - XX4)

= (A + 2)m-det((A + 2- jfe)In - A)

D


k X As_Spec r =




2 t - 4 i - 4 - 2


Additional results




3 1 - 2 s

Spec O3 = 1 g



A2 + (c - o)A + (c - fc)I = cJ













13










d lt 2 2k V log2n



ln(n-l)d lt

Cycles and cuts






y = [yiy2---ynt





defined by









D(1) 0 00 Dlt2gt 0

0 0 D^c


Cycles and cuts 25






















Cycles and cuts 27







Q = DD = A - A




Additional Results








fo(x) = 1 fx(x)=x f2(x) = x2-k

fq(x) - xfq-xx) ~(k- l) -2(i) (q gt 3)




9ix) = 0(a) + fix) + + fi(x)

Cycles and cuts 29















on S((fgt)







mour 1981)
























r bull




CT 1c=

bulllm-n+1




X-ir


JX = (Uy UN) bull








Dw = 0 Cz = 0


[ wj-1w = z =






Additional Results













z = w + n Dw = 0 Cz = 0















R(xyV) gt R(xyT)


6

The tree-number










The tree-number 39



adj Q = K(F)J


det(D[)det(Dpound)





6(D = n-2det (J + Q)


(nl - J)(J + Q) = nJ + nQ - J2 - JQ = nQ







det (J + Q)J = n2Ki





mi ms-i

then


r=l






x(L(r)2k-2) = (2k)m-nx(Tk)


The tree-number 41







K(T) =n-2aTcn)






bull



aKa- n) = (-l)ax(Ka a - 1 - fj) = - a)01


K(Kaua2a) = n-2(n)(n - a^11 (n)(n - CL)0-1

= n ~2 (n - ai)ai-1 (n - a




Additional Results


lt --t I N nmdash1

1 I nkn n mdash 1



n - l





1 2h~1


nMh) = ~[(2 + v3) + (2 - v3)h] + h


The tree-number 43









(n - M)ltr(FcM) = (-1)














r(A) =n~(No + Ne) == Ne (mod 2)




det A(K4) = 3(-l)22deg + 3(-l)321 = - 3


X(F A) - Xn + cxA1 + c2A-2 + + cn






C4 = na - 2rib






We shall write


















XY













Additional Results














(rA) pound ( r A )








aij=Eiej (lltijltn)





8












X(F z) = zn + c2zn~2 + CiZ1- + = z6pz2)



A - f deg JA ~ [J 0



























uT) lt 1 + Amax(r)



bull







P QlQ R j










t














Additional Results




ltrgt s



Mi = min i(Qx)ux=O


vweE v





Mi lt SX ( 2





= 1


2(2r23) k



problem (QP)



1 gtw(r) - n2



PART TWO

Colouring problems

9






bull




Hence

C(Kn u) = uu - l)(w - 2) (u - n + 1)
























are given by

mi(T) =





D


+ 3U(2) + U(i)) O (u(3) + 3U(2) + U

6U(5) + llU(4) + 6W(3) + W(2)

M5 + 36w4 - 75u3 + 78u2 - 31u





C(cT u) = uC(F u - 1)

C(sT u) = uu - 1)C(F u - 2) + uC(T u - 1)



j(w - l)(j) = uC(Tu - 1)








C(Fu) =



C(Ta)






Additional Results


C(Wn u) = u(u - 2)71-1 -(- ( - l ) - 1 ^ - 2)






C(Lh u) = (u2 -3u + 3)h + (u- 1)(3 - u)h + (1 - u)h) + u2 -

C(Mh u) = (u2 -3u + 3)h + (u - 1)(3 - u)h - (1 - u)h - 1






u8 -24u7+260u6 - 1670u5+6999w4 - 19698u3+36408u2 -40240u+20170




-u(u - l)(w - 2) ] T Ci(l - u)


1 10 56 230 759 2112 5104 10912 20880 35972

55768 77152 93538 96396 80572 50808 21302 4412


k

Pk = ^ m r f c ( r ) r=0


2fc=0 V r = 0 fc=0 V




fe








mdash n~5

C(F (3 bull










Given a graph F let




10

Subgraph expansions



R(Txy) = Er




L 81 78 36 9 1J






QET- 01 as follows

_ f 1






C(Tu)=SCET




c(rlaquo)= X) E




|VT VS + c = n - VS + c = n- rS)

D





j




M = C(Tu) = unR(T -u~ -1)



bx = - 9 62 = 36 h = -84 + 9 = -75 b4 = 117 - 45 + 6 = 78

65 = -81 + 78 - 36 + 9 - 1 = -31


3t3 u)=u6 - 9M5 + 36M4 - 75u3 + 78u2


(1) (-l)lt6-=(7) for i = 0lg-2

(2) ( - l ) raquo - V i = ( f l - i ) - -



















Additional Results
















pound a - 1)SCEV U





11








X(T) - ]T YS)SCET






Y(S)r = Y(S1)r1Y(S2)r2-



S1CE1 S2CE2



ICJ








X(S) = X(S1)X(S2)


SCET



52 (-i)|S2|iog x(s2



XT)= n YS)SCEF




We shall prove that

iogx(r)=SCET







bull





bull
















Additional Results



1 - (1 - u)r-n - (1 - u)s~n - (1 - it)- + (2 - u)(l - u)~n

(1 - (1 - u)r-n)(l - (1 - u)s~n)(l - (1 - uf-n)





7 1 2

2 +











Explicitly we have





12





Q(Cnu) = q(Cnu) =



















Q(F u) = 1 + (30u~n+l + l U - n +



Q(K4u)


and so

C(K4 u) = u4- 6u3 + llu2 - au + b


C(K4 u) = u4 - 6w3 + llu2 - 6u = u(u - l)(u - 2)u - 3)







A Aa A A5



6 l j


A2 A3 A4

C u ( u - l ) u(u - l)(u2 - 3w + 3)

Q (u - l)u u(u - 2)(u - I)2 u(u2 -3u + 3)(u - I)3


= u(u - l)(u - 2)(w2 - 3w + 3)





(I)6

= u6 - 9u5 + 34u4 - 67u3 + 67w2






) = P(u)Q(Tu)



P(u) =C((V1DV2u)














Additional Results



where ) = (-ir-1-(T1)


r(K2u) = 1u

r(K3u)= -lu-lf

r(K4 u) = -(2u - 3)u(u - 2)2

r(C4u)= l (w- l ) 3 -


r(K5 u) = -(6u4 - 48w3 + 140u2 - 176u + 81)(u - l)4(u - 3)4

r(W5 u) = (3u2 - 9u + 7)u(u - 2f(u2 - 3u + 3)

r(W-u) = (2u2 - 6u + 5)u(u - 2)(M2 - 3u + 3)2

r(K23 u) = (u3 - 6u2 + llu - 7)u(u2 - Zu + 3)3

r(C5u)= -l(u-l)



5681012

15

1024612

112410


14 19 3 130 15 10 1








2 + v ^ - 4 u + 8)lt u


+ x7 + 3xs + 4x9 + 3x10 Ux 12

- 91a15 - 261a16 - 290x17

+ 24a13

254x18

8z14

13













r(X n(X)) = r(X) s(X M(X)) = s(X) - 1








AXA = X = xix2)xt



gtlUxi x a_i









X+=XUXeuro X~=XXL









A - ^ B

VC - ^ V












ni = W 6 C | r(W) = r0 - t |W~| = j |











cients

k x 7 i










Additional Results





r 0 36 84 75 35 9 136 168 171 65 10120 240 105 15180 170 30170 70114 1256216

L l










r(xya) =x-1



A







TYxy)=TTyx)


fcgt0








14




C(T u) = (-l)-1^ ]T laquo(1 - )t=i



C(Tu) =unR(F-u-1-l)






n-2






bull








0(1^) = 0(T2)



C(rl) = -C(r ( e ) l )













^ n-l-A ^ (n~l-l



Additional Results









0(Kn) = (n - 2) (n gt 2)

0(Lh) = 2fe - ft - 1 (ft gt 3)

0(Mh) = 2h - ft (ft gt 2)




C(Tu)=uC(ru)









C ( F (u + y - 1 ) 2 ) = ] pound ( V c




log TB(n)log TAn)


PART THREE


15













r D














duv) = dau)av))






Additional Results























16





G Gv = GGV = |VT|








l (12)

(123)

(132) (23)




hkGET=gth~lkefl

= (gh)~lgk euro Q




a


































Additional Results
























17

Symmetric graphs











Pg-t=O0-

[a]


























Thus we have




















Additional Results



Jjy LtyLiU

jLjyyj J-JW


GV lt k(k - 1)LVW


G lt 2nGv lt 2nn (n - 1) = 2 (n)2











18











111 = 2 ( 0 lt t lt t - l )

|Ft| = 3x2-1



9 = 9i ^o = 9T192 xi=g~lxag (i = 1 2 )


bull bull [a]












x0 = (34) Xl = (12) x2 = (35) x3 = (14)



5 (i

8 7


51 = (1234)(5678) g2 = (123785)(46)

xo - (36)(45) X = (16)(47) x2 =













as claimed D


[XiXj] = (XiXj)2

















1 [xoxx][xo OM

= [XoXx]







| ( laquo - l ) lt A lt | ( t + 2)




2)2

= (x5x0x5)2 = x5xlx5 = 1








|Aut(fi)| = |AutS6| = 1440 = 30 x 3 x 24




x3 = |(o6)(c)(de)| x4 = |(c)|


F5 = 54xZ2 F4 = D8x Z2 F3 = (Z2)3

F2 = (Z2)2 F=Z2


Additional Results




^(a - d)(b - c)






n = ^rp(p2 - 1) when p = 79 (mod 16)

n = ^p2ipA - 1) when p = 351113 (mod 16)









a2 = 1 (era)2 = 1 (ab)2 = 1 (a^b)2 = 1 abaa2 = b2


a2 = 1 aaf = 1 (crb)2 = 1 (a1))2 = 1 ( a~V) 2 = 1

(a-363)2 = 1 a4b4a4 = ba a4baab = ba3b







Ft = (a-ibii= 12t)





t Fl F2 F3 F4 F5

1 Z 3

2 Z 2 S3

3 Z 2 (Z2)2 D12

4 Z2 (Z2)2 As2 (Z2)2

)5 Z2 (Z2)2










a = (12) (34675) (89X) b = (1246853)(79X) a = (12)(34)(56)(9JsT)


19









r f






KST

ST -1+ K




(K9)(KV) ~ (Kg(n)g(v))
















V - U0Ui U[ = V








bull



Additional Results






ltfgt(vw)=je







20











D





Shij = Shi(uv)











C i Cj

a0 ai a bo bi bj


1 2 3t(Q3) = 0 0 0 0

3 2 1








f 1 ) ( I ktKn) ~ 0 n - 2 i(Kkik) = 0 0 0


1 4

0 t-2 2 - 82f - 4 t - 3


















-Imdash 1


kh bd-i 1 c 2 ca









(A - AI)(Ao + Ai + + Ad) = 0





d

j=o


rs mdash Shi(vrvs)





Additional Results




21 - 121 - 221 - 2 1 + 11 + 11 1122 1 - 11 - 1

2121 - 121 - 1 I + 11 + 1 1122 I - 11 - 1









s(t + 1) (s - a + l)t 1 a(t + 1)




^CA I K A 1 A fe j




21















TO 1k d i C2

h a2 bull

bull bull cd

bull adA








vo() = 1










BK = (BK)( - KB



D





^) ioltiltd)


3=0











n3




0 1 02r r 40 r - 1 2r - 4




v0 =

12r

gt(r-l) Vi =

1r - 21 - r

v2 = 1 - 21

m(Ai) n(ui

m(A2) =

1 + (r - 2)22r + (1 - r)2rr - 1)

1 i r ( r _ 1 N = 2~(r ~ 1)(r(u2v2) 1-



_ _ 16 4 -2s



Additional Results



d(q-l)(d-l)(q-l)(q-l) 12 d




i) =m




n(0 lt i lt d)



=E-(r = 0



B mdash090

108

045


0810

1512




n lt -m(m + 3)













22

Imprimitivity

















Imprimitivity 175






















Imprimitivity 177

















Additional Results



i(I) = 521 125 i(D) = 32111 11123



Imprimitivity 179












(O6)H


+ (15)(23)(46)- (12)(36)(45)- (14)(25)(36)-gt (16)(23)(45)- (15)(26)(34)

(ac)i-

(laquo)-(ce) H

-gt (14)(26)(35)-raquo (16)(25)(34)^ (16)(24)(35)- (13)(25)(46)- (12)(35)(46)

(ad)v-(6c) H(bf)-(c)( e ) -

bull+ (13)(24)(56)-gt (12)(34)(56)- (13)(26)(45)- (15)(24)(36)- (14)(23)(56)


23




no(kg) = l + k + k(k-l) + + k(k- l)^g-3l


fcfc-lfc-lfc-lllll



k k ~ 1 k 1 K 1 1 1 1 1 kj







fcl and fcfc-lljfc






rok

10

J f c - 110- 1

10

fc-1c



() cud-i(X) + (k - c - )ud) = 0




(1 lt i lt d)








k 2gcosa (j = 12 d)







2





F = 2h-(h2 + l)cos2a G = (h2 - 1)sin2a



)



nk (4h-2



















m _ (fc + fc3)(4fc-4-A2)W (Jfe-A)(6-2 + 5A)


s5 + s4 + 6s3 - 2s2 + (9 - mi)s - 15 = 0







m = k = -(A3 - fc2 + fc)








lt2qx - ( 2





we have





R(X) = gm(X)nk = (4ft - X2)(k - X)(f + A)



(R(X) - 1)A2 + R(X)(f - k)X - (R(X)fk - Ah) = 0











Additional Results


















REFERENCES


192 References


References 193



194 References





References 195




196 References



References 197


198 References


References 199


200 References




References 201



Index






Index 203















Paley graph 129

204 Index










walk 9

Index 205


ALGEBRAIC GRAPH THEORY - cloudflare-ipfs.com

Documents

Transcript of ALGEBRAIC GRAPH THEORY - cloudflare-ipfs.com