operativa 2 lect1

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CHAPTER 2

DEFINITIONS AND EXAMPLES

In Chapter 1 you saw how a graph can be used to depict the relationships between cer-tain objects; you simply represent the objects by venices, and the reiationships by edgesjoiaing the vertices. In order to investigate such relationships more deepiy, we need tostudy the theory of graphs in greater detail. To this end, we now inrroduce some usefuiterminology which will be needed in what follows.

2.1 ADJACENCY AND INCIDENCE

Since graph theory is primariiy concerned with interconnections between objects, we

shall need some terminology which teils us when certain vertices and edges occur nextto each other in a graph. This terminology applies to all graphs, and can be used equaiiyweli for wires connecting terminals in an electrical network, bonds connecring atoms ina chemicai molecule, or roads connecting towns on a road map.

DEFINITIONS. Let v and w be vertic'e s of a graph. If v and w are joined by an edgee, then v and w are said to be ad,jacent. Moreol,er, v and w are said ¡o áe incident withe, and e is said ¡o áe incident with v and w.

31



CHAPTER 2, DEFINITIONS AND EXAMPLES

v and w ate adjacent;

' :i ;"il:,:ií : i::t'-:'

You have already seen two ways of representing a graph-as a diasram consisting of

points joined by iines. and as a set of vertices and a list of edges. The pictorialrepresenution is useful in many situations, especially when we wish to examine thestructure of the graph as a whole, but its value diminishes as soon as we need to describelarge or complicated graphs. For example, if we wish to store a large _eraph in a computer,then a pictorial representation may well be unsuitable, and some other method wouldthen be necessary.

One possibility is to store the set of vertices and the list of edges; this method is oftenused in practice, especially when the graph is 'sparse'-that is, it has a lot of vertices butrelatively few edges. Another method is to take each vertex in turn and list those verticesthat are adjacent to it. By joining each vertex to its 'neighbors', we can easiiy reconstructthe graph. Ye¡ another method is to draw up a table indicating the pairs of vertices thatare adjacent, or a table indicating'¡hich vertices are incidenr wirh which edges.

Each of these methods has its advantases, but the last one is particularly useful in a

number of practicai appiications. In this method, we represent a graph bv a rectangularrr-ra.v of numbers, called a matrix: a mairix with trows and /coiumns is callecl akxtntan'ix. Such matrices lend themselves to mechanical manipulation, and in severalapplications of graph theory they yield the most natural wav of formuiaring a givenprobiem. There are various types of matrices thar can be used to specify a given graph.Here we describe the most important ones-the adjacency matrix and the incidencemarrix. For sirnplicity, we restrict our attention to graphs without loops.

The Adjacency Matr¡x

Consider the following exampie

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tr4r':, row 1

row 2

row 3

row 4

col. col. coi. col.

1231i*yt0 1 o t\1 0 t ,l0 1 o 1l! 2 1 al

(Jir the lefi-hand side we have a graph withT'bur vertices, and on the righrhand side rvehave a I x 4 matri.r. The nt¡mbers appearing in the matrix refer to ¡he number of edgesjoining the corresponding vertices in the graph. For example.

F



2.1 AOJACENCY AND INCIDENCE

vertices I and 2 arejoined by 1 edge, so 1 appears in row 1 column 2 and inrow 2 coiumn I

vertices 2 and 4 arejoined by 2 edges, so 2 appears in row 2 column 4 and inrow 4 column 2

vertices I and 3 arejoined by 0 edges, so 0 appears in row I column 3 and inrow 3 column 1

Note that every entry on the main (top-left to bottom-righ| diagonal is 0, since the graphhas no loops. Note also that the matrix is symmetrical about this main diagonal.

We can generalize this idea as follows.

DEFINITION. Let G be a graph without loops, with n vertices labeled I ,2 3 ,...,n.Ilre adjacency matrix M(G) is the nxn matrix in which the entry in row i and column jis the number of edges joining the vert¡ces i and j.

The lncidence MatrixWhereas the adjacency matrix of a graph involves the adjacency of vertices, the

incidence matrix involves the incidence of vertices and edges. To see what is involved,consider the foilowing example; we have circled the iabels of the vertices to distinguishthem from the labels of the edges.

'w"col. col. co[. col. col. col.123456+++t++

ro,,O*lt o o 1 o o\,o*@*f 1 1 0 0 I 1

I,o*(D*|i0 I I 0 0 0l,o*O--\o o 1 i 1 tl

On the left-hand side we have a graph with/our vertices and s¡x edges, and on the righr-hand side we have a 4 x 6 matrix. Each of the numbers appearing in the matrix is either 1

or 0, depending on whether or not the corresponding vertex and edge are incident witheach other. For example,

vertex O is incident with edge 4, so I appears in row I coiumn 4

vertex @ is not incident with edge 4. so 0 appears in row 2 column 4

We can generalize this idea, as follows.

DEFINITION. Let G be a graph without loops, with n vertices labeled @,@,@,..., n

and m edges labeled I,2.3,..., m.The incidence matrix I(G) is the nxm matix in whichthe enrry in row i and column j is L if vertex i is incident with edge j, and 0 otherwise.

Note that the incidence matrix depends on the particular way that thé vertices and

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CHAPTER 2. DEFINIT}ONS AND EXAMPLES

edges are labeleci. We obtain one incidence matrix from another ltv interchanging rows(correspondiltg to relabeiing the venices) and columns (corresponding to relabeiing theeciges ).

2.2 PATHS AND CYCLES

Many of the aoplications of -eraph theory involve 'getting from one vertex to anottler' ina graph. For example, how can you find the shortest route between one London Under-ground station and another? Other examples inciude the routing of a telephone call be-tween one subscriber and another, the flow of current between two terminals of anelecrical netu'ork, and the uacing of a maze. Our aim in this section is to make this ideaprecise by means of some definitions. We srart by defining awalkin a graph.

DBFINITION. A walk of length kin a graphG is a succession ofkedges of G of theJbrm

Ul''\'tW'tVX""' )"i

We denorc this walk by uvt',x...';2. and refer to it as a walk between ¿¡ and:.

No¡e that tlle 'second veriex' of each edge is the same as the 't-irsr vertex' of the next.Iuturti'rr-1,v. \¡'e c?n think of this as a walk tiom ¿¿ io v, tben ro w. then ¡o,v, and so on, untilwe eventuaily end up ai vertex r. Alternatively, since the edges have no specifieddirec¡ion. we can think of it as a *'alk from : to y and so on, eventually, to r. r,v. v, and u.

Thus we can alsr¡ denate iiris waik by :1..-nvvrr, and refer to it as a walk t¡etween z andu^

We do not require ail rhe edges or vertices in a walk to be differenl For exampie, inthe fbliowing graph

Lwrr't.vw'.t::, iE av,alk o! lenzth 9 between l and y, which includes the edge i4r iwice, ¿udlhe ver¡ices v-, tt,, y, and z twic.-. This leads to the lbllowing r1efinitions.

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ü

L

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2.3 EXAMPLES OF GRAPHS

DEFINITIONS. If all the edges (but not ne('essarily all the verrices) of a v,alk aredifferent , then the walk is called a trail. If in addition, all the vertices are differenr, the nthe trail is called a path.

In the above diagram, the waik yz4-wxy ís atrail which is not a path (since the verticesy and : both occur twice), whereas the waik vn,r]: has no repeated vertices, and istherefore a path. Note that a walk such as u:vuz ts also a trail, as long as the twooccunences of uz refer to the two different edges joining u and z.

It is also useful to have special terms for those waiks or trails which stan and finishat the same vertex.

DEFINITIONS. A closed walk in a graph G is a succession of edges of G of the form

L¿v, vw, wx, ..., yz, zu

If all of these edges are dffirent, then the walk is cálled a closed trail. If in addition, theyertices u,v-,wyr.,...,!,2 are all different, then t.he trail is called a cycle.

In the above graph, the closed walk ¿¿vwyvzuis aclosed trail which is not a cycle (sincethe vertex v occurs twice), whereas ¡he closed trails zz, vwxJv, and vwryzv are all cycles.A cycle of length three, such as rlryv or wxyw, is called a triangle, for obvious reasons.In describing closed walks, we can ailow any vertex to be the stafiing vertex. For exampie,the triangle ),14/yu can equally well be described by the letters wyvw or yvwy or (since thedirection is immaterial) by ywvy, vywv, or wvyw.

We can use the concept of a path to explain exactiy what is meant by a connectedgraph. Recall from Chapter I that a graph is connected if it is 'in one piece'. For example,the tbilowing graph is not connected, but can be split into four connected pieces.

The observation that there is a path between;r and y (which lie in the same piece). but notbetween u end y (which lie in different pieces), ieads to the following definitions.

DEFII,{ITIONS. A graph G is connected if there is a path inG betvveen anv given pairofvertices, a¿d disconnected othenvise. Every disconnec'ted graph can be split up into¿¡ number of connected suhgraphs , cal/ed components.


We now introduce some irnportant tvpes of graphs. You shouid make sure ¡'ou are famii-iar wiih them since they will appear liequentiy. both in applicarions and as examples.

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CHAPTER 2. DEFINITIONS AND EXAMPLES

Complete Graphs

A complete graph is a gr aph in which every two distinct vertices are joined by exactlyone edge. The complete graph with n vertices is denoted by Xo. Note ihat, apafi from Ka,we usually draw the vertices of K" in the form of a regular polygon.

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Note that N" is regular of degree 0.

Cycle Graphs

A c¡:t:le graph is a graph consisting of a single cycle. The cycle glaph with n verdces is

denoted by C,.

€---\i

cr cz

Noe that C" is reguku of degree 2. and has n edges.

Path Graphs

A path graph rs a graph consisting of a siigle path. The path $aph wiih n venices is de-noted by P,.

,Ar$.'f-,.,M;

The graph K" is regular of degree n - 1, and therefore has Lrn(n - l) edges, by conse-

quence 3 of the handshaking lemma.

NullGraphsA null graph is a graph conuining no edges. The null graph with ¡? vertices is denotedbyN,.

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Note that Pnhas n -1 edges, and can be obtained from the cycle graph C" by removingany ed_se.

Bipartite Graphs

Of particular importance in applications are the bipartite graphs. A bipartite graph is a

graph whose vertex-set can be spiit into sets A and B in such a way that each edge of thegraph joins a vertex in A to a vertex in B. We can distinguish the vertices in A from thosein B by drawing the former in black and the latter in white. so that each edge is incidentwith a black vertex and a white vertex. Some examples of bipartite graphs are

A/ \ black

f vertices

whi¡e,- e u 'Ivertices-_-- I

A complete bipartite graph is a bipartite graph in which each black vertex is joined toeach white vertex by exactly one edge. The complete bipartite graph with r black verti-ces and s white vefiices is denoted by K",. A complete bipartite graph of the form K,, iscalled a star graph. Some examples of complete bipartite graphs are

*n+"'Mffi"'Mr.s Kz.z Kzt Kz;

Note that K,., has r + s venices (r vertices of degree s, and s vertices of degree r), and rsedges. Note also that K,., = K,.,: it is usual, but not necessary, to put the smaller of r and s

t'irst.

Cube Graphs

Of particular interest among the bipartite graphs are the cube graphs. These graphs haveimportant applications in coding theory, and may be constructed by taking as vertices ailbinarywords(sequencesof 0sand ls)of agivenlengthandjoiningtwoof thesevenicesif the corresponding binary words differ in just one place. The graph obtained in this

way from the binary words of length fr is cailed the t-cube (or k-dimensional cube), andis denoted by p¡.



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The Petersen Graph

our next example is a famous graph which has already appeared in the problems ofChapter 1. It is known as the Petersen graph, and has several interesting propertieswhich you will discover as you progress through the book. The petersen graph may bedrawn in various ways, two of which are shown here

Julius Petersen (1839-1910) was a Danish mathematician, who discussed the graphnamed after him in a paper of 1898.

Trees

A connected graph which contains no cycles is called a tree. Some examples of trees areas foilows; the tree on the right is particularly well known for its barkl

? t -1 ''rr' { p¡-*-*H r-r--------a-__-_. D--_.-

¿ ¿ r\ ¿ b ib ..

If G isa

connected graph, then a spanning tree in G is a subgraph of G which includesevery vertex of G and is aiso a tree. For e.xampie. a graph and three of its spanning trees are

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J-_]J---¡zyxa graph 6

fBt1l-_+--¡:y.x

spanning tree

twl*_7,/ /\¿ J \:yx

spanning tree

rp!t

tV.:y.¡spanning tree

F

I!,iFEI

The number of spanning trees in a graph can be very large. For example, the petersengraph has no fewer than 2000 different spanning trees.

Unions and ComplementsThere are ,"u"rui operations we can perform on graphs in order to form new ones. Thesimplest of these is to form their union, which is the graph whose componerts are ¡he



CHAPTER 2. DEFINITIONS AND EMMPLES

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individual graphs. For example, the null graph iV, is the union of n copies of N,, and thegraph

AANis the union of two copies of Kr, two copies of K3, and one copy of K2.r.

Finally, if G is a simple graph, we form its complement G by taking the vertex-set ofG anci joining two vertices by an edge whenever they are not joined in G. For example,the complement of K, is Nn, and the complement of

-thatis, the path graph Pr. Note that if we uke the complement of G, rhen we get back

to the original graph G.

PROBLEMS

Here, and throughout this book, the dagger (i) indicates chalLenge problems.

Adiacency and lncidenceQz.l Consider the following graph:

Classify each of the following statements as ffue or false:a. u and z are adjacent; e. u is incident with c;

b. v anci z are adjacent; f. á is incident with d;

c. d and z are adjacenr; g. ¿ is incident with w;d. v is incident with d; h. v is incident withl

IIrs

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PROBLEMS

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Consider the following graph:

*---l\\¡ I/t \\ t.

'#Classify each of the following statements as true or false:a. e is incident with u; c. a is incident with e;

b. c is adjacent to x; d. w is adjacent to v.

Match up each of the following graphs with its adjacency matrix and incidence

matrix:

(b)

Adjacency matrices.

lncidence matrices.

('ili:r)2.4. Write down the adjacency matrices of rhe graphs

t--T\,L-V,t(a)

(l :i;il (iiii) (i ilri)

(r i:riil fi;:iiilD,

o(a)

l2

i--Al,/ \l ')vv*,or'



CHAPTEN 2. DEF¡NITIONS AND EXAMPLES

Draw the graphs whose adjacency matrices are

62.6- what can you say about the sum of ihe numbers in any rorv or corumn of anadjacency matrix?

a2'7 ' The foliowing diagrams illustrate a graph with three different labelings. Findthe adjacency matrix in each case, and exprain the connections between thesethree matrices.

Write down the incidence maftices of rhe following graphs:

o(a)

o2.9. Draw ¡he graph whose incidence ma¡rix is

io l l o o o\lr o o r o oi

{;lr¿;;l0 0 0 0 0 I

I

\o o o o r o/

(b)

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)1 i I I 0 0 0 0r

li r o ü i r o o\l{) o ü o o o o ol\o o o ¡ o I r rl\0 0 I 0 I 0 t t/



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PROBLEMS

2.L0. What can you say about the sum of the numbers in

a. any row of an incidence matrix?

b. any column of an incidence matrix?

2.11.i Consider the foilowing graph:

Write down the adjacency matrix M of this graph, and compute the matrices M2

andM3. What does the entry in row i and columni of each of these matricesrepresent? Make a guess as to what the entries of M¿ represent, and prove yourresult.

Z.lZ.t (For those who have studied linear algebra.) The eigenvalues of a simpiegraph are defined as the eigenvaiues of its adjacency matrix.

a. Use the results of the previous problem to prove that

i. the sum ofthe eigenvalues ofG is zero;

ii. the sum of their squares is 2m, where m is the number of edges of G;

iii. the sum of their cubes is 6r, where r is the number of triangles in G.

b. Show that the eigenvalues of K, are -l (n - I times), and n ' I '

c. What are ¡he eigenvalues of K,,?

Paths and Cycles42.13. Compiete the statements conceming the graph shown.

a. xyzzry is a

-

of length between and

-;b. vuvzv is a

-of

length between and

-;c. vw is a of iength between and

-;

d- uvwxt¡zu is a of lensth between and



In the tbllowing graph, finda. a rvalk of length 7 between r and w,;

b. cycles of lengths I , 2, 3, and 4;

c. a path crf maximum length.

CHAPTER 2. DEFINITIONS AND EXAMPLES

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2.15. Find all the paths between s and z in the graph

2.T6. In the following graph, finda. a ciosed walk which is not a closed trail;

b. a closed trail which is not a cycle;

c. all the cycles of lengths l, 2, 3, and 4.

Examples of Graphs02.17, Draw ihe tbllowing graphs:

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F

1t

Lt-t-i;i

{ii

a. K¡;

b. ,V*;

c.C*;d.Pr;

e'Kr.rlf.K¡,i



PNABLEMS

02.18. Fill in the following table:

Ke ce Qrtetra- l

hedron r cub€

trta-hedron

dodeca-hedron

rcog-hed¡or Petersn

number of

vertices

number ofedges

degree ofeach vertex

2.L9. Complete the following statements:

a. the graph K.., is a regular $aph only when

-;b. the oniy bipartite Platonic gmph is the

-;

c. the graph K,., is ttre union of

-

and

-;d. if G is a simple graph with /? verrices wbich is regular of degree r, then G isregular of degree

-;e. if G is a simple graph with r? vertices and m edges, then G has vertices

and

-

edges.

2.20. From the set ofgraph cards, locate the cards that depict the graphs

a. No;

b' Kt;

c. Cu;

d' Pt;

2.21 Which of the tbllowing graph cards depict brpanite graphs?

e. K¡,51 i: C o;

f. K1; i. Po.

g. the tenahedron graph:

h. the union of K. and K.;

a. number 38;

b. number 78;

c. number 106;

d. number 128:

e. number i29:f. numbe¡ 130:

g. number 152;

h. number 154

2.22 Using your graph cards, determine the number of connected bipartite graphs

with five vertices.

02.23. Show that in any bipartite graph all cycles have even lengü.

Z.ZA.i fnV complete tripartite graph K..,, consists of three sets of vertices of sizes r.

s, and f, with edges joining two vertices if and only if they iie in different sets.

a. Draw the graphs K,.... and K3¡,3.

b. How many veriices and edges has K,,.'?

c. What is the compiement of K.,,.,?

2.25. The girth of a graph is the length of i.s shoftest cycle. Find the girths of (a) Ks;(b) Kosi (c) the Petersen graph; (d) the Platonic graphs: (e) the 6-cube Qo.



CHAPTEN 2, DEFINITIONS AND EMMPLES

2.26. The circumference of a graph is the length of its longest cycie. Find the cir-cumferences of (a) Kn: (b) Ks.s; (c) the Petersen graph; (d) the dodecahedrongraph; (e) the 4-cube Q*.

2.27 .+ How many dift'erent cycies of lengch 5 does the Petersen graph have?

2.28. Locate the graph cards depicting the complement of each of the followinggraphs:

a. the path graph P,;

b. the complete bipartite graph K.*.2.29. How many components has each of the follorving graphs?

(a) Ns; (b) K¡,¡; (c) Rr.s.

02.30. Prove that a graph and its complement cannot both be disconnected.

2.3I. A graph is called self-complementary if it is isomorphic to its complement.

a. Show that C, is self-complementary, and find two other self-complementarygraphs.

b. Prol'e that a self-complernentary graph has 4k or 4k + 1 vertices. tbr someinteger ft.

4232. There are 14 trees with six or fewer vertices. Draw them, and locate rhe graphcards depicting them.

2.33. There are l1 trees with seven vertices. Drarv them.0.234, Show that if a tree has n vertices then it has n - 1 edges.

2.35. Use ¡he handshak;ng iemma to prove that every tree rvirh two or rnore verticeshas at least two vertices of degree i.

7.3(). Shorv that every tree is a bioartite graph.

ü2.37. Find ajl the spannins irees in each oi the ibllo..,ring graphs:

E

!\

F

ir

i

s

¡,

i,

1{?r. .4I \B// i

Lx'. i

,,\ED

{¿¡

"!I

r--7t'!i.,' Iv\.?C

( b,t

1.38. Drew some (but nor ail !) of rhe 2000 spanning trees of rhe Petersen graph.

?..39.i I'he line graoh t (G ) of a sin'iole graph G is rhe graph obrarned by taking theedge.s of G as vertices, and joining trvo of r"hese vel-lices whene'¡er the corre-sponding edges of G have a vertex in common. Find an expression for the num-ber of edges of L(G) in terms of the degrees ol the 'zeriices of G, and show that:

3. if C" is the c)'cle graph wirh ¡r verrices, thc.n l(C" i is isomor.ohic to C";

b. t(:{") has lntn - 1) vertic'es, and is regular of rlegree 2n - 4',

c. l(tetrahedron) = octahedroud. ihe contplement of I(Kr) is rhe Petersen graph.

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operativa 2 lect1

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