Research Article An Algebraic Representation of Graphs and...

Hindawi Publishing CorporationInternational Journal of CombinatoricsVolume 2013 Article ID 347613 14 pageshttpdxdoiorg1011552013347613

Research ArticleAn Algebraic Representation of Graphs and Applications toGraph Enumeration

Acircngela Mestre

Centro de Estruturas Lineares e Combinatorias Universidade de Lisboa Av Prof Gama Pinto 2 1649-003 Lisboa Portugal

Correspondence should be addressed to Angela Mestre mestreciifculpt

Received 23 July 2012 Accepted 25 September 2012

Academic Editor Xueliang Li

Copyright copy 2013 Angela MestreThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We give a recursion formula to generate all the equivalence classes of connected graphs with coefficients given by the inversesof the orders of their groups of automorphisms We use an algebraic graph representation to apply the result to the enumerationof connected graphs all of whose biconnected components have the same number of vertices and edges The proof uses Abelrsquosbinomial theorem and generalizes Dziobekrsquos induction proof of Cayleyrsquos formula

1 Introduction

As pointed out in [1] generating graphs may be useful fornumerous reasons These include giving more insight intoenumerative problems or the study of some properties ofgraphs Problems of graph generation may also suggest con-jectures or point out counterexamples The use of generatingfunctions (or functionals) in the enumeration or generationof graphs is standard practice both in mathematics andphysics [2ndash4] However this is by no means obligatory sinceany method of manipulating graphs may be used

Furthermore the problem of generating graphs takinginto account their symmetries was considered as early asthe 19th century [5] and more recently for instance in [6]In particular in quantum field theory generated graphs areweighted by scalars given by the inverses of the orders of theirgroups of automorphisms [4] In [7 8] this was handled fortrees and connected multigraphs (with multiple edges andloops allowed) on the level of the symmetric algebra on thevector space of time-ordered field operators The underlyingstructure is an algebraic graph representation subsequentlydeveloped in [9] In this representation graphs are associatedwith tensorswhose indices correspond to the vertex numbersIn the former papers this made it possible to derive recursionformulas to produce larger graphs from smaller ones byincreasing by 1 the number of their vertices or the numberof their edges An interesting property of these formulas is

that of satisfying alternative recurrences which relate eithera tree or connected multigraph on 119899 vertices with all pairsof their connected subgraphs with total number of verticesequal to 119899 In the case of trees the algorithmic description ofthe corresponding formula is about the same as that used byDziobek in his induction proof of Cayleyrsquos formula [10 11]Accordingly the formula induces a recurrence for 119899119899minus2119899that is the sum of the inverses of the orders of the groupsof automorphisms of all the equivalence classes of trees on 119899

vertices [12 page 209]For simplicity here by graphs we mean simple graphs

However our results generalize straightforwardly to graphswith multiple edges allowed One instance of an algorithmfor finding the biconnected components of a connectedgraph is given in [13] Our goal here is rather to generate allthe equivalence classes of connected graphs so that they aredecomposed into their biconnected components and have thecoefficients announced in the abstract To this end we givea suitable graph transformation to produce larger connectedgraphs from smaller ones by increasing the number of theirbiconnected components by one unit This mapping is thenused to extend the recurrence of [7] to connected graphsThis new recurrence decomposes the graphs into theirbiconnected components and in addition can be generalizedto restricted classes of connected graphs with specifiedbiconnected components The proof proceeds as suggestedin [7] That is given an arbitrary equivalence class whose

2 International Journal of Combinatorics

representative is a graph on 119898 edges say 119866 we show thatevery one of the119898 edges of the graph119866 adds 1(119898sdot |aut119866|) tothe coefficient of 119866 To this end we use the fact that labeledvertices are held fixed under any automorphism

Moreover in the algebraic representation framework theresult yields a recurrence to generate linear combinations oftensors over the rational numbers Each tensor represents aconnected graph As required these linear combinations havethe property that the sum of the coefficients of all the tensorsrepresenting isomorphic graphs is the inverse of the orderof their group of automorphisms In this context tensorsrepresenting generated graphs are factorized into tensorsrepresenting their biconnected components As in [7ndash9] akey feature of this result is its close relation to the algorithmicdescription of the computations involved Indeed it is easyto read off from this scheme not only algorithms to performthe computations but even data structures relevant for animplementation

Furthermore we prove that when we only considerthe restricted class of connected graphs whose biconnectedcomponents all have say 119901 vertices and 119903 edges the cor-responding recurrence has an alternative expression whichrelates connected graphs on 120583 biconnected components withall the 119901-tuples of their connected subgraphs with totalnumber of biconnected components equal to 120583 minus 1 Thisinduces a recurrence for the sum of the inverses of the ordersof the groups of automorphisms of all the equivalence classesof connected graphs on 120583 biconnected components with thatpropertyThe proof uses an identity related to Abelrsquos binomialtheorem [14 15] and generalizes Dziobekrsquos induction proof ofCayleyrsquos formula [10]

This paper is organized as follows Section 2 reviews thebasic concepts of graph theory underlyingmuch of the paperSection 3 contains the definitions of the elementary graphtransformations to be used in the following Section 4 gives arecursion formula for generating all the equivalence classes ofconnected graphs in terms of their biconnected componentsSections 5 and 6 review the algebraic representation andsome of the linear mappings introduced in [7 9] Section 7derives an algebraic expression for the recurrence of Section4 and for the particular case in which graphs are such thattheir biconnected components are all graphs on the samevertex and edge numbers An alternative formulation for thelatter is also given Finally Section 8 proves a Cayley-typeformula for graphs of that kind

2 Basics

We briefly review the basic concepts of graph theory that arerelevant for the following sectionsMore detailsmay be foundin any standard textbook on graph theory such as [16]

Let 119860 and 119861 denote sets By [119860]2 we denote the set of allthe 2-element subsets of 119860 Also by 2119860 we denote the powerset of 119860 that is the set of all the subsets of 119860 By card119860 wedenote the cardinality of the set 119860 Furthermore we recallthat the symmetric difference of the sets 119860 and 119861 is given by119860Δ119861 = (119860 cup 119861) (119860 cap 119861)

Here a graph is a pair 119866 = (119881 119864) where 119881 sub N is afinite set and 119864 sube [119881]

2 Thus the elements of 119864 are 2-element

subsets of 119881 The elements of 119881 and 119864 are called verticesand edges respectively In the following the vertex set of agraph 119866 will often be referred to as 119881(119866) the edge set as119864(119866) The cardinality of119881(119866) is called the order of 119866 writtenas |119866| A vertex 119907 is said to be incident with an edge 119890 if119907 isin 119890 Then 119890 is an edge at 119907 The two vertices incidentwith an edge are its endvertices Moreover the degree of avertex 119907 is the number of edges at 119907 Two vertices 119907 and 119906

are said to be adjacent if 119907 119906 isin 119864 If all the vertices of 119866are pairwise adjacent then 119866 is said to be complete A graph119866lowast is called a subgraph of a graph 119866 if 119881(119866lowast) sube 119881(119866) and

119864(119866lowast

) sube 119864(119866) A path is a graph 119875 on 119899 ge 2 vertices suchthat 119864(119875) = 119907

1 1199072 119907

2 1199073 119907

119899minus1 119907119899 119907

119895isin 119881(119875) for

all 119895 = 1 119899 The vertices 1199071and 119907

119899have degree 1 while

the vertices 1199072 119907

119899minus1have degree 2 In this context the

vertices 1199071and 119907

119899are linked by 119875 and called the endpoint

verticesThe vertices 1199072 119907

119899minus1are called the inner vertices

A cycle is a graph 119862 on 119899 gt 2 vertices such that 119864(119862) =

1199071 1199072 119907

2 1199073 119907

119899minus1 119907119899 119907

119899 1199071 119907

119895isin 119881(119862) for all

119895 = 1 119899 every vertex having degree 2 A graph issaid to be connected if every pair of vertices is linked bya path Otherwise it is disconnected Given a graph 119866 amaximal connected subgraph of 119866 is called a component of119866 Furthermore given a connected graph a vertex whoseremoval (together with its incident edges) disconnects thegraph is called a cutvertex A graph that remains connectedafter erasing any vertex (together with incident edges) (respany edge) is said to be 2-connected (resp 2-edge connected)A 2-connected graph (resp 2-edge connected graph) is alsocalled biconnected (resp edge-biconnected) Given a con-nected graph119867 a biconnected component of119867 is a maximalsubset of edges such that the induced subgraph is biconnected(see [17 Section 64] for instance) Here we consider that anisolated vertex is by convention a biconnected graphwith nobiconnected components

Moreover given a graph 119866 the set 2119864(119866) is a vector spaceover the field Z

2such that vector addition is given by the

symmetric difference The cycle space C(119866) of the graph 119866

is defined as the subspace of 2119864(119866) generated by all the cyclesin 119866 The dimension ofC(119866) is called the cyclomatic numberof the graph119866 We recall that dimC(119866) = card119864(119866)minus |119866|+ 119888where 119888 denotes the number of connected components of thegraph 119866 [18]

We now introduce a definition of labeled graph Let 119871be a finite set Here a labeling of a graph 119866 is a mapping119897 119881(119866) rarr 2

119871 such that cup119907isin119881(119866)

119897(119907) = 119871 and 119897(119907) cap 119897(1199071015840

) = 0

for all 119907 1199071015840 isin 119881(119866) with 119907 = 1199071015840 In this context 119871 is called

a label set while the graph 119866 is said to be labeled with 119871 orsimply a labeled graph In the sequel a labeling of a graph 119866

will be referred to as 119897119866 Moreover an unlabeled graph is one

labeled with the empty setFurthermore an isomorphism between two graphs 119866 and

119866lowast is a bijection 120593 119881(119866) rarr 119881(119866

lowast

) which satisfies the fol-lowing conditions

(i) 119907 1199071015840 isin 119864(119866) if and only if 120593(119907) 120593(1199071015840) isin 119864(119866lowast

)

(ii) 119871 cap 119897119866(119907) = 119871 cap 119897

119866lowast(120593(119907))

International Journal of Combinatorics 3

Clearly an isomorphism defines an equivalence relation ongraphs In particular an isomorphism of a graph119866 onto itselfis called an automorphism (or symmetry) of 119866

3 Elementary Graph Transformations

We introduce the basic graph transformations to change thenumber of biconnected components of a connected graph byone unit

Here given an arbitrary set 119883 let Q119883 denote the freevector space on the set119883 overQ the set of rational numbersAlso for all integers 119899 ge 1 and 119896 ge 0 and label sets 119871 let

119881119899119896119871

= 119866 |119866| = 119899 dimC (119866) = 119896 119866 is labeled

by 119897119866 119881 (119866) 997888rarr 2

119871

(1)

Furthermore let(i) 119881119899119896119871conn = 119866 isin 119881

119899119896119871

119866 is connected(ii) 119881119899119896119871biconn = 119866 isin 119881

119899119896119871

119866 is biconnectedIn what follows when 119871 = 0 we will omit 119871 from the upperindices in the previous definitions

We proceed to the definition of the elementary linearmappings to be used in the followingNote that for simplicityour notation does not distinguish between two mappingsdefined both according to one of the following definitionsone on 119881

119899119896119871 and the other on 1198811199011199021198711015840

with 119899 = 119901 or 119896 = 119902 or119871 = 119871

1015840 This convention will often be used in the rest of thepaper for all themappings given in this sectionTherefore wewill specify the domain of the mappings whenever confusionmay arise

(i) Adding a biconnected component to a connected graphlet 119871 be a label set Let 119866 be a graph in 119881

119899119896119871

conn Let119881(119866) = 119907

119894119894=1119899

For all 119894 = 1 119899 let K119894denote

the set of biconnected components of 119866 such that119907119894isin 119881(119867) for all 119867 isin K

119894 Let L denote the set

of all the ordered partitions of the set K119894into 119901 dis-

joint sets L = 119911119894= (K

(1)

119894 K

(119901)

119894) cup

119901

119897=1K

(119897)

119894

= K119894and K

(119897)

119894cap K

(1198971015840

)

119894= 0 forall119897 119897

1015840

= 1 119901 with 119897 = 119897

1015840

Furthermore let J denote the set ofall the ordered partitions of the set 119897

119866(119907119894) into 119901

disjoint sets J = 119908119894= (119897

119866(119907119894)(1)

119897119866(119907119894)(119901)

)

cup119901

119897=1119897119866(119907119894)(119897)

= 119897119866(119907119894) and 119897

119866(119907119894)(119897)

cap 119897119866(119907119894)(1198971015840

)

= 0 forall119897

1198971015840

= 1 119901 with 119897 = 1198971015840

Finally let 119866 be a graph in119881119901119902

biconn such that 119881(119866) cap 119881(119866) = 0 (In case the graph119866 does not satisfy that property we consider a graph1198661015840 instead such that1198661015840 cong 119866 and119881(119866)cap119881(1198661015840) = 0We

will not point this out explicitly in the following) Let119881(119866) = 119906

119897119897=1119901

In this context for all 119894 = 1 119899define

119903

119866

119894 Q119881

119899119896119871

conn 997888rarr Q119881119899+119901minus1119896+119902119871

conn

119866 997891997888rarr sum

119911119894isinL119908

119894isinJ

119866119911119894

119908119894

(2)

where the graphs 119866119911119894119908119894

satisfy the following

(a) 119881(119866119911119894119908119894

) = 119881(119866) 119907119894 cup 119881(119866)

(b) 119864(119866119911119894119908119894

) = 119864(119866) cup 119909 119910 isin 119864(119866) 119907119894notin 119909 119910

119901

⋃

119897=1

119909 119906119897 119909 119907

119894 isin 119864 (119866) 119909 isin cup

119867lowastisinK(119897)

119894

119881 (119867lowast

) (3)

(c) 119897119866119911119894

119908119894

|119881(119866)119907

119894= 119897

119866|119881(119866)119907

119894and 119897

119866119911119894

119908119894

(119906119897) = 119897

119866(119907119894)(119897)

for all 119897 = 1 119901

The mappings 119903119866119894are extended to all of Q119881

119899119896119871

conn bylinearity For instance Figure 1 shows the result ofapplying the mapping 1199031198624

119894to the cutvertex of a 2-edge

connected graph with two biconnected componentswhere 119862

4denotes a cycle on four vertices

Furthermore let 119883 sube 119881119901119902

biconn Given a linearcombination of graphs 120599 = sum

119866isin119883120572119866119866 where 120572

119866isin Q

we define

119903120599

119894= sum

119866isin119883

120572119866119903119866

119894 (4)

We proceed to generalize the edge contraction oper-ation given in [16] to the operation of contracting abiconnected component of a connected graph

(ii) Contracting a biconnected component of a connectedgraph let 119871 be a label set Let 119866 be a graph in 119881

119899119896119871

conn Let 119866 isin 119881

1199011199021198711015840

biconn be a biconnected component of 119866where 1198711015840 = cup

119907isin119881(119866)119897119866(119907) Define

119888119866 119881

119899119896119871

conn 997888rarr 119881119899minus119901+1119896minus119902119871

conn 119866 997891997888rarr 119866lowast

(5)

where the graph 119866lowast satisfies the following

(a) 119881(119866lowast) = 119881(119866)119881(119866)cup119907 where 119907 = min 1199071015840

isin

N 1199071015840

notin 119881(119866) 119881(119866)(b) 119864(119866lowast) = 119909 119910 isin 119864(119866) 119909 119910 cap 119864(119866) = 0

cup 119909 119907 119909 119910 isin 119864 (119866) 119864 (119866) and 119910 isin 119881 (119866) (6)

(c) 119897119866lowast |119881(119866)119881(

119866)

= 119897119866|119881(119866)119881(

119866)

and 119897119866lowast(119907) =

cup119906isin119881(

119866)119897119866(119906)

For instance Figure 2 shows the result of applying themapping 119888

1198624

to a 2-edge connected graph with threebiconnected componentsWe now introduce the following auxiliary mappingLet 119871 be a label set Let 119866 be a graph in 119881

119899119896119871 Let119881(119866) = 119907

119894119894=1119899

Let 1198711015840 be a label set such that119871cap119871

1015840

= 0 Also letI denote the set of all the orderedpartitions of the set 1198711015840 into 119899 disjoint setsI = 119910 =

(1198711015840(1)

1198711015840(119899)

) cup119899

119895=11198711015840(119895)

= 1198711015840 and 119871

1015840(119894)

cap 1198711015840(119895)

=

0 forall119894 119895 = 1 119899 with 119894 = 119895 In this context define

1205851198711015840 Q119881

119899119896119871

997888rarr Q119881119899119896119871cup119871

1015840

119866 997891997888rarr sum

119910isinI

119866119910 (7)


119903cutvertex

= 4 + 8 + 4)(

Figure 1 The linear combination of graphs obtained by applying the mapping 1199031198624119894

to the cutvertex of the graph consisting of two trianglessharing one vertex

=)(119888

Figure 2The graph obtained by applying themapping 1198881198624

to a graphwith three biconnected components

where the graphs 119866119910satisfy the following

(a) 119881(119866119910) = 119881(119866)

(b) 119864(119866119910) = 119864(119866)

(c) 119897119866119910

(119907119894) = 119897

119866(119907119894) cup 119871

1015840(119894) for all 119894 = 1 119899 and119907119894isin 119881(119866)

The mapping 1205851198711015840 is extended to all of Q119881

119899119896119871 by line-arity

4 Generating Connected Graphs

We give a recursion formula to generate all the equivalenceclasses of connected graphs The formula depends on thevertex and cyclomatic numbers and produces larger graphsfrom smaller ones by increasing the number of their bicon-nected components by one unitHere graphs having the sameparameters are algebraically represented by linear combina-tions over coefficients from the rational numbers The keyfeature is that the sum of the coefficients of all the graphs inthe same equivalence class is given by the inverse of the orderof their group of automorphisms Moreover the generatedgraphs are automatically decomposed into their biconnectedcomponents

In the rest of the paper we often use the followingnotation given a group119867 by |119867| we denote the order of119867Given a graph 119866 by aut119866 we denote the group of automor-phisms of 119866 Accordingly given an equivalence class A byautAwe denote the group of automorphisms of all the graphsin A Furthermore given a set 119882 sube 119881

119899119896119871 by E(119882) wedenote the set of equivalence classes of all the graphs in119882

We proceed to generalize the recursion formula forgenerating trees given in [7] to arbitrary connected graphs

Theorem 1 For all 119901 gt 1 and 119902 ge 0 suppose that 120573119901119902119887119894119888119900119899119899

=

sum1198661015840isin119881119901119902

119887119894119888119900119899119899

1205901198661015840119866

1015840 with 1205901198661015840 isin Q is such that for any equivalence

classA isin E(119881119901119902

119887119894119888119900119899119899) the following holds (i) there exists1198661015840 isin A

such that 1205901198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 = 1|autA| In this context

given a label set 119871 for all 119899 ge 1 and 119896 ge 0 define 120573119899119896119871119888119900119899119899

isin

Q119881119899119896119871

119888119900119899119899by the following recursion relation

12057310119871

119888119900119899119899= 119866 119908ℎ119890119903119890 119866 = (1 0) 119897

119866(1) = 119871

1205731119896119871

119888119900119899119899= 0 119894119891 119896 gt 0

120573119899119896119871

119888119900119899119899=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1) 119903

120573119901119902

119887119894119888119900119899119899

119894(120573

119899minus119901+1119896minus119902119871

119888119900119899119899))

(8)

Then 120573119899119896119871119888119900119899119899

= sum119866isin119881119899119896119871

119888119900119899119899

120572119866119866 with 120572

119866isin Q Moreover for any

equivalence class C isin E(119881119899119896119871119888119900119899119899

) the following holds (i) thereexists 119866 isin C such that 120572

119866gt 0 (ii) sum

119866isinC 120572119866= 1|autC|

Proof Theproof is very analogous to the one given in [7] (seealso [8 19])

Lemma 2 Let 119899 ge 1 and 119896 ge 0 be fixed integers Let 119871 be alabel set Let 120573119899119896119871conn = sum

119866isin119881119899119896119871

conn120572119866119866 be defined by formula (8)

Let C isin E(119881119899119896119871conn ) denote any equivalence class Then thereexists 119866 isin C such that 120572

119866gt 0

Proof The proof proceeds by induction on the number ofbiconnected components 120583 Clearly the statement is true for120583 = 0We assume the statement to hold for all the equivalenceclasses in E(119881119899minus119901+1119896minus119902119871conn ) with 119901 = 2 119899 minus 1 and 119902 =

0 119896 whose elements have 120583minus1 biconnected componentsNow suppose that the elements of C isin E(119881119899119896119871conn ) have 120583

biconnected components Let120573119901119902biconn = sum1198661015840isin119881119901119902

biconn1205901198661015840119866

1015840 with

1205901198661015840 isin Q Recall that by (4) the mappings 119903120573

119901119902

biconn119894

read as

119903

120573119901119902

biconn119894

= sum

1198661015840isin119881119901119902

biconn

12059011986610158401199031198661015840

119894 (9)

Let 119866 denote any graph in C We proceed to show that agraph isomorphic to119866 is generated by applying themappings119903

120573119901119902

biconn119894

to a graph 119866lowast

isin 119881119899minus119901+1119896minus119902119871

conn with 120583 minus 1 biconnectedcomponents and such that 120584

119866lowast gt 0 where 120584

119866lowast is the

coefficient of119866lowast in 120573119899minus119901+1119896minus119902119871conn Let119866 isin 1198811199011199021198711015840

biconn be any bicon-nected component of the graph 119866 where 1198711015840 = cup

119907isin119881(119866)119897119866(119907)


Contracting the graph119866 to the vertex 119906 = min1199061015840 isin N 1199061015840

notin

119881(119866) 119881(119866) yields a graph 119888119866(119866) isin 119881

119899minus119901+1119896minus119902119871

conn with 120583 minus 1

biconnected components Let D isin E(119881119899minus119901+1119896minus119902119871conn ) denotethe equivalence class such that 119888

119866(119866) isin D By the inductive

assumption there exists a graph 119867lowast

isin D such that 119867lowast

cong

119888119866(119866) and 120584

119867lowast gt 0 Let 119907

119895isin 119881(119867

lowast

) be the vertex mapped to 119906of 119888

119866(119866) by an isomorphism Relabeling the graph 119866 with the

empty set yields a graph 1198661015840

isin 119881119901119902

biconn Applying the mapping1199031198661015840

119895to the graph119867lowast yields a linear combination of graphs one

of which is isomorphic to 119866 That is there exists119867 cong 119866 suchthat 120572

119867gt 0

Lemma 3 Let 119899 ge 1 and 119896 ge 0 be fixed integers Let 119871 bea label set such that card119871 ge 119899 Let 120573119899119896119871

119888119900119899119899= sum

119866isin119881119899119896119871

119888119900119899119899

120572119866119866 be

defined by formula (8) Let C isin E(119881119899119896119871119888119900119899119899

) be an equivalenceclass such that 119897

119866(119907) = 0 for all 119907 isin 119881(119866) and 119866 isin C Then

sum119866isinC 120572

119866= 1


0 119896 whose elements have 120583minus 1 biconnected componentsand the property that no vertex is labeled with the empty setNow suppose that the elements of C isin E(119881119899119896119871conn ) have 120583

biconnected components By Lemma 2 there exists a graph119866 isin C such that 120572

119866gt 0 where 120572

119866is the coefficient of 119866

in 120573119899119896119871

conn Let 119898 = card119864(119866) Therefore 119898 = 119896 + 119899 minus 1 Byassumption 119897

119866(119907) = 0 for all 119907 isin 119881(119866) so that |autC| = 1

We proceed to show thatsum119866isinC 120572

119866= 1 To this end we check

from which graphs with 120583 minus 1 biconnected components theelements ofC are generated by the recursion formula (8) andhow many times they are generated

Choose any one of the 120583 biconnected components of thegraph 119866 isin C Let this be the graph 119866 isin 119881

1199011199021198711015840

biconn where1198711015840

= cup119907isin119881(

119866)119897119866(119907) Let 1198981015840

= card119864(119866) so that 1198981015840

=

119902 + 119901 minus 1 Contracting the graph 119866 to the vertex 119906 with119906 = min1199061015840 isin N 119906

1015840

notin 119881(119866) 119881(119866) yields a graph119888119866(119866) isin 119881

119899minus119901+1119896minus119902119871

conn with 120583 minus 1 biconnected components LetD isin E(119881119899minus119901+1119896minus119902119871conn ) denote the equivalence class containing119888119866(119866) Since 119897

119888119866(119866)

(119907) = 0 for all 119907 isin 119881(119888119866(119866)) we also have

|autD| = 1 Let 120573119899minus119901+1119896minus119902119871conn = sum119866lowastisin119881119899minus119901+1119896minus119902119871

conn120584119866lowast119866

lowast with120584119866lowast isin Q By Lemma 2 there exists a graph 119867 isin D such

that 119867 cong 119888119866(119866) and 120584

119867gt 0 By the inductive assumption

sum119866lowastisinD 120584

119866lowast = 1 Now let 119907

119895isin 119881(119867) be the vertex which is

mapped to 119906 of 119888119866(119866) by an isomorphism Let 1198661015840 isin 119881

119901119902

biconnbe the biconnected graph obtained by relabeling 119866 with theempty set Also let A isin E(119881

119901119902

biconn) denote the equivalenceclass such that1198661015840 isin A Apply themapping 119903119866

1015840

119895to the graph119867

Note that every one of the graphs in the linear combination1199031198661015840

119895(119867) corresponds to a way of labeling the graph 119866

1015840 with

119897119888119866(119866)

(119906) = 1198711015840 Therefore there are |autA| graphs in 119903

1198661015840

119895(119867)

which are isomorphic to the graph 119866 Since none of the

vertices of the graph 119867 is labeled with the empty set themapping 119903

1198661015840

119895produces a graph isomorphic to 119866 from the

graph 119867 with coefficient 120572lowast119866

= 120584119867

gt 0 Now formula (8)prescribes to apply the mappings 119903

1198661015840

119894to the vertex which

is mapped to 119906 by an isomorphism of every graph in theequivalence classD occurring in 120573

119899minus119901+1119896minus119902119871

conn (with non-zerocoefficient) Therefore

sum

119866isinC

120572lowast

119866= |autA| sdot sum

119866lowastisinD

120584119866lowast

= |autA|

(10)

where the factor |autA| on the right hand side of the firstequality is due to the fact that every graph in the equivalenceclass D generates |autA| graphs in C Hence according toformulas (8) and (4) the contribution to sum

119866isinC 120572119866is 1198981015840

119898Distributing this factor between the 119898

1015840 edges of the graph119866 yields 1119898 for each edge Repeating the same argumentfor every biconnected component of the graph 119866 proves thatevery one of the119898 edges of the graph119866 adds 1119898 tosum

119866isinC 120572119866

Hence the overall contribution is exactly 1 This completesthe proof

We now show that 120573119899119896119871conn satisfies the following property

Lemma 4 Let 119899 ge 1 and 119896 ge 0 be fixed integers Let 119871 and 1198711015840

be label sets such that 119871 cap 1198711015840

= 0 Then 120573119899119896119871cup1198711015840

119888119900119899119899= 120585

1198711015840(120573

119899119896119871

119888119900119899119899)

Proof Let 1205851198711015840 119881

119899119896119871

rarr 119881119899119896119871cup119871

1015840

and 1205851198711015840 119881

119899minus119901+1119896minus119902119871

rarr

119881119899minus119901+1119896minus119902119871cup119871

1015840

be defined as in Section 3The identity followsby noting that 120585

1198711015840 ∘ 119903

120573119901119902

biconn119894

= 119903

120573119901119902

biconn119894

∘1205851198711015840

Lemma 5 Let 119899 ge 1 and 119896 ge 0 be fixed integers Let 119871 bea label set Let 120573119899119896119871

119888119900119899119899= sum

119866isin119881119899119896119871

119888119900119899119899

120572119866119866 be defined by formula

(8) Let C isin E(119881119899119896119871119888119900119899119899

) denote any equivalence class Thensum119866isinC 120572

119866= 1|aut (C)|

Proof Let 119866 be a graph in C If 119897119866(119907) = 0 for all 119907 isin 119881(119866)

we simply recall Lemma 3 Thus we may assume that thereexists a set1198811015840 sube 119881(119866) such that 119897

119866(119881

1015840

) = 0 Let 1198711015840 be a labelset such that 119871 cap 119871

1015840

= 0 and card 1198711015840 = card1198811015840 Relabelingthe graph 119866 with 119871 cup 119871

1015840 via the mapping 119897 119881(119866) rarr 2119871cup1198711015840

such that 119897|119881(119866)119881

1015840 = 119897119866|119881(119866)119881

1015840 and 119897(119907) = 0 for all 119907 isin 1198811015840

yields a graph 1198661015840

isin 119881119899119896119871cup119871

1015840

conn such that |aut1198661015840| = 1 LetD isin E(119881119899119896119871cup119871

1015840

conn ) be the equivalence class such that 1198661015840 isin DLet 120573119899119896119871cup119871

1015840

conn = sum1198661015840isin119881119899119896119871cup119871

1015840

conn1205841198661015840119866

1015840 with 1205841198661015840 isin Q By Lemma

3 sum1198661015840isinD 120584

1198661015840 = 1 Suppose now that there are exactly 119879 dis-

tinct labelings 119897119895 119881(119866) rarr 2

119871cup1198711015840

119895 = 1 119879 such that119897119895|119881(119866)119881

1015840 = 119897119866|119881(119866)119881

1015840 119897119895(119907) = 0 for all 119907 isin 119881

1015840 and 119866119895isin D

where119866119895is the graph obtained by relabeling119866with 119897

119895 Clearly

120584119866119895

= 120572119866gt 0 where 120584

119866119895

is the coefficient of 119866119895in 120573

119899119896119871cup1198711015840

conn Define 119891

119895 119866 997891rarr 119866

119895for all 119895 = 1 119879 Now repeating the


12057310conn =

12057320conn =

12057330conn =

12057340conn =

12057341conn =

12057331conn =

12057340conn =

1

2

1

3

1

8

+1

2

+1

2

+1

4

+1

3

1

2

1

2

1

2

Figure 3 The result of computing all the pairwise non-isomorphic connected graphs as contributions to 120573119899119896

conn via formula (8) up to order119899 + 119896 le 5 The coefficients in front of graphs are the inverses of the orders of their groups of automorphisms

same procedure for every graph inC and recalling Lemma 4we obtain

sum

1198661015840isinD

1205841198661015840 =

119879

sum

119895=1

sum

119866isinC

120584119891119895(119866)

= 119879sum

119866isinC

120572119866= 1 (11)

That is sum119866isinC 120572

119866= 1119879 Since |autC| = 119879 we obtain

sum119866isinC 120572

119866= 1|aut (C)|

This completes the proof of Theorem 1

Figure 3 shows 120573119899119896conn for 1 le 119899 + 119896 le 5 Now givena connected graph 119866 let V

119866denote the set of biconnected

components of119866 Given a set119883 sub cup119899

119901=2cup119896

119902=0119881119901119902

biconn let119881119899119896

119883=

119866 isin 119881119899119896

conn E(V119866) sube E(119883) with the convention 119881

10

119883=

(1 0) With this notation Theorem 1 specializes straight-forwardly to graphs with specified biconnected components

Corollary 6 For all 119901 gt 1 and 119902 ge 0 suppose that 120574119901119902119887119894119888119900119899119899

=

sum1198661015840isin119883119901119902 120590

1198661015840119866

1015840 with 1205901198661015840 isin Q and 119883

119901119902

sube 119881119901119902

119887119894119888119900119899119899 is such that

for any equivalence class A isin E(119883119901119902

) the following holds(i) there exists 1198661015840 isin A such that 120590

1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =

1|autA| In this context for all 119899 ge 1 and 119896 ge 0 define 120574119899119896119888119900119899119899

isin

Q119881119899119896


12057410

119888119900119899119899= 119866 119908ℎ119890119903119890 119866 = (1 0)

1205741119896

119888119900119899119899= 0 if 119896 gt 0

120574119899119896

119888119900119899119899=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1) 119903

120574119901119902

119887119894119888119900119899119899

119894(120574

119899minus119901+1119896minus119902

conn ))

(12)

Then 120574119899119896119888119900119899119899

= sum119866isin119881119899119896

119883

120572119866119866 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896

119902=0119883119901119902 Moreover for any equivalence class C isin

E(119881119899119896119883

) the following holds (i) there exists 119866 isin C such that120572119866gt 0 (ii) sum

119866isinC 120572119866= 1| autC|

Proof The result follows from the linearity of the mappings119903119866

119894and the fact that larger graphs whose biconnected com-

ponents are all in119883 can only be produced from smaller oneswith the same property

5 Algebraic Representation of Graphs

We represent graphs by tensors whose indices correspondto the vertex numbers Our description is essentially that of[7 9] From the present section on we will only considerunlabeled graphs

Let 119881 be a vector space over Q Let S(119881) denote thesymmetric algebra on 119881 Then S(119881) = ⨁

infin

119896=0S119896(119881) where

S0(119881) = Q1 S1(119881) = 119881 and S119896(119881) is generated by the freecommutative product of 119896 elements of 119881 Also let S(119881)otimes119899denote the 119899-fold tensor product of S(119881) with itself Recallthat the multiplication in S(119881)

otimes119899 is given by the compon-entwise product

sdot S(119881)otimes119899

times S(119881)otimes119899

997888rarr S(119881)otimes119899

(1199041otimes sdot sdot sdot otimes 119904

119899 1199041015840

1otimes sdot sdot sdot otimes 119904

1015840

119899)

997891997888rarr 11990411199041015840


1198991199041015840

119899

(13)

where 119904119894 1199041015840119895denote monomials on the elements of 119881 for all

119894 119895 = 1 119899 We may now proceed to the correspondencebetween graphs on 1 119899 and some elements of S(119881)otimes119899


1

(a)

1 2

(b)

3

21

(c)

Figure 4 (a) An isolated vertex represented by 1 isin S(119881) (b) A 2-vertex tree represented by 119877

12isin S(119881)

otimes2 (c) A triangle representedby 119877

12sdot 119877

13sdot 119877

23isin S(119881)

otimes3

First for all 119894 119895 = 1 119899 with 119894 = 119895 we define the followingtensors in S(119881)

otimes119899

119877119894119895= 1

otimes119894minus1

otimes 119907 otimes 1otimes119895minus119894minus1

otimes 119907 otimes 1otimes119899minus119895

(14)

where 119907 is any vector different from zero (As in Section3 for simplicity our notation does not distinguish betweenelements say 119877

119894119895isin S(119881)

otimes119899 and 119877119894119895

isin S(119881)otimes1198991015840

with 119899 = 1198991015840

This convention will often be used in the rest of the paperfor all the elements of the algebraic representationThereforewe will specify the set containing consider the given elementswhenever necessary) Now for all 119894 119895 = 1 119899 with 119894 = 119895 let

(i) a tensor factor in the 119894th position correspond to thevertex 119894 of a graph on 1 119899

(ii) a tensor 119877119894119895

isin S(119881)otimes119899 correspond to the edge 119894 119895 of

a graph on 1 119899

In this context given a graph 119866 with 119881(119866) = 1 119899

and 119864(119866) = 119894119896 119895119896119896=1119898

we define the following algebraicrepresentation of graphs

(a) If119898 = 0 then119866 is represented by the tensor 1otimessdot sdot sdototimes1 isin

S(119881)otimes119899

(b) If 119898 gt 0 then in S(119881)otimes119899 the graph 119866 yields a

monomial on the tensors which represent the edgesof 119866 More precisely since for all 1 le 119896 le 119898 eachtensor 119877

119894119896119895119896

isin S(119881)otimes119899 represents an edge of the graph

119866 this is uniquely represented by the following tensor119878119866

1119899isin S(119881)

otimes119899 given by the componentwise productof the tensors 119877

119894119896119895119896

119878119866

1119899=

119898

prod

119896=1

119877119894119896119895119896

(15)

Figure 4 shows some examples of this correspondence Fur-thermore let 119883 sube 1 119899

1015840

be a set of cardinality 119899 with1198991015840

ge 119899 Also let 120590 1 119899 rarr 119883 be a bijection With thisnotation define the following elements of S(119881)otimes119899

1015840

119878119866

120590(1)120590(119899)=

119898

prod

119896=1

119877120590(119894119896)120590(119895119896) (16)

In terms of graphs the tensor 119878119866120590(1)120590(119899)

represents a discon-nected graph say 1198661015840 on the set 1 1198991015840 consisting of a

1

3 4

2

(a)

5

3

4

1

2

(b)

Figure 5 (a) The graph represented by the tensor 1198781198661234

isin S(119881)otimes4

(b) The graph represented by the tensor 1198781198661542

isin S(119881)otimes5 associated

with 119866 and the bijection 120590 1 2 3 4 rarr 1 2 4 5 1 997891rarr 1 2 997891rarr

5 3 997891rarr 4 4 997891rarr 2

graph isomorphic to119866whose vertex set is119883 and 1198991015840minus119899 isolatedvertices in 1 119899

1015840

119883 Figure 5 shows an exampleFurthermore let T(S(119881)) denote the tensor algebra on

the graded vector space S(119881) T(S(119881)) = ⨁infin

119896=0S(119881)

otimes119896 InT(S(119881)) the multiplication

∙ T (S (119881)) times T (S (119881)) 997888rarr T (S (119881)) (17)

is given by concatenation of tensors (eg see [20ndash22])


119899) ∙ (119904

1015840


1015840

1198991015840)

= 1199041otimes sdot sdot sdot otimes 119904

119899otimes 119904

1015840


1015840

1198991015840

(18)

where 119904119894 1199041015840

119895denote monomials on the elements of 119881 for all

119894 = 1 119899 and 119895 = 1 1198991015840 We proceed to generalize the

definition of the multiplication ∙ to any two positions of thetensor factors Let 120591 S(119881)otimes2 rarr S(119881)otimes2 119904

1otimes 119904

2997891rarr 119904

2otimes 119904

1

Moreover define 120591119896= 1

otimes119896minus1

otimes120591otimes1otimes119899minus119896minus1

S(119881)otimes119899 rarr S(119881)otimes119899for all 1 le 119896 le 119899 minus 1 In this context for all 1 le 119894 le 1198991 le 119895 le 119899

1015840 we define ∙119894119895

S(119881)otimes119899 times S(119881)otimes1198991015840

rarr S(119881)otimes119899+1198991015840

bythe following equation


119899) ∙119894119895(1199041015840


1015840

1198991015840)

= (120591119899minus1

∘ sdot sdot sdot ∘ 120591119894) (119904


119899)

otimes (1205911∘ sdot sdot sdot ∘ 120591

119895minus1) (119904

1015840


1015840

1198991015840)



119899otimes 119904

119894

otimes 1199041015840

119895otimes 119904

1015840

1otimes sdot sdot sdot otimes

1199041015840


1015840

1198991015840

(19)

where 119904119894(resp 1199041015840

119895) means that 119904

119894(resp 1199041015840

119895) is excluded from the

sequence In terms of the tensors 1198781198661119899

isin S(119881)otimes119899 and 119878119866

1015840

11198991015840 isin

S(119881)otimes1198991015840

the equation previous yields

119878119866

1119899∙1198941198951198781198661015840

11198991015840 = 119878

119866

120590(1)120590(119899)sdot 1198781198661015840

1205901015840(1)120590

1015840(1198991015840) (20)

where 119878119866120590(1)120590(119899)

1198781198661015840

1205901015840(1)120590

1015840(1198991015840)isin S(119881)

otimes119899+1198991015840

120590(119896) = 119896 if 1 le 119896 lt

119894 120590(119894) = 119899 120590(119896) = 119896 minus 1 if 119894 lt 119896 le 119899 and 1205901015840

(119896) = 119896 + 119899 + 1

if 1 le 119896 lt 119895 1205901015840(119895) = 119899 + 1 1205901015840(119896) = 119896 + 119899 if 119895 lt 119896 le 1198991015840


1

3 41

32 2119866prime119866

(a)

5

6

1

4 3

2

119867

(b)

Figure 6 (a)The graphs represented by the tensors 1198781198661234


and 1198781198661015840

123isin S(119881)

otimes3 (b) The graph represented by the tensor 11987811986716

=

119878119866

1234321198781198661015840

123isin S(119881)

otimes6

Clearly the tensor 1198781198661119899

∙1198941198951198781198661015840

11198991015840 represents a disconnected

graph Now let sdot119894= 1

otimes119894minus1

otimes sdot otimes 1otimes119899minus119894minus1

S(119881)otimes119899

rarr S(119881)otimes119899minus1

In T(S(119881)) for all 1 le 119894 le 119899 1 le 119895 le 1198991015840 define the following

nonassociative and noncommutative multiplication

119894119895= sdot

119899∘ ∙

119894119895 S(119881)

otimes119899

times S(119881)otimes1198991015840

997888rarr S(119881)otimes119899+1198991015840

minus1

(21)

The tensor 1198781198661119899

1198941198951198781198661015840

11198991015840 represents the graph on 119899 + 119899

1015840

minus 1

vertices say119867 obtained by gluing the vertex 119894 of the graph 119866to the vertex 119895 of the graph1198661015840 If both119866 and1198661015840 are connectedthe vertex 119899 is clearly a cutvertex of the graph 119867 Figure 6shows an example

6 Linear Mappings

We recall some of the linear mappings given in [9]LetB

119899119896sub S(119881)

otimes119899 denote the vector space of all the ten-sors representing biconnected graphs in 119881

119899119896

biconn Let B =

B10⨁B

20⨁

infin

119899=3⨁

119896max(119899)

119896=1B

119899119896sub T(S(119881)) where 119896max(119899) =

(119899(119899 minus 3)2) + 1 is by Kirchhoff rsquos lemma [18] the maximumcyclomatic number of a biconnected graph on 119899 vertices Let

the mapping Δ B rarr T(S(119881)) be given by the followingequations

Δ (1) = 1 otimes 1

Δ (119861119866

1119899) =

1

119899

119899

sum

119894=1

Δ119894(119861

119866

1119899) if 119899 gt 1

(22)

where 119866 denotes a biconnected graph on 119899 vertices repre-sented by 119861

119866

1119899isin S(119881)

otimes119899 To define the mappings Δ119894 we

introduce the following bijections

(i) 120590119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894

119895 + 1 if 119894 + 1 le 119895 le 119899

(ii) 120584119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894 minus 1

119895 + 1 if 119894 le 119895 le 119899

(23)

In this context for all 119894 = 1 119899 with 119899 gt 1 the mappingsΔ119894 S(119881)

otimes119899

rarr S(119881)otimes119899+1 are defined by the following

equation

Δ119894(119861

119866

1119899) = 119861

119866

120590119894(1)120590

119894(119899)

+ 119861119866

120584119894(1)120584

119894(119899)

= 119861119866

1119894+1119894+2119899+1+ 119861

119866

1119894119894+1119899+1

(24)

where (resp 119894 + 1) means that the index 119894 (resp 119894 + 1)is excluded from the sequence The tensor 119861119866

1119894119899+1(resp

119861119866

1119894+1119894+2119899+1

) is constructed from 119861119866

1119899by transferring the

monomial on the elements of119881which occupies the 119896th tensorfactor to the (119896 + 1)th position for all 119894 le 119896 le 119899 (resp 119894 + 1 le

119896 le 119899) Furthermore suppose that 119861119866120587(1)120587(119899)

isin S(119881)otimes1198991015840

andthat the bijection 120587 is such that 119894 notin 120587(1 119899) sub 1 119899

1015840

In this context define

Δ119894(119861

119866

120587(1)120587(119899)) = 119861

119866

120584119894(120587(1))120584

119894(120587(119899))

(25)

in agreement withΔ(1) = 1otimes1 It is straightforward to verifythat the mappings Δ

119894satisfy the following property

Δ119894∘ Δ

119894= Δ

119894+1∘ Δ

119894 (26)

where we used the same notation for Δ119894

S(119881)otimes119899

rarr

S(119881)otimes119899+1 on the right of either side of the previous equation

and Δ119894 S(119881)

otimes119899+1

rarr S(119881)otimes119899+2 as the leftmost operator

on the left hand side of the equation Accordingly Δ119894+1

S(119881)otimes119899+1

rarr S(119881)otimes119899+2 as the leftmost operator on the right

hand side of the equationNow for all 119898 gt 0 define the 119898th iterate of Δ

119894 Δ119898

119894

S(119881)otimes119899

rarr S(119881)otimes119899+119898 recursively as follows

Δ1

119894= Δ

119894 (27)

Δ119898

119894= Δ

119894∘ Δ

119898minus1

119894 (28)

where Δ119894

S(119881)otimes119899+119898minus1

rarr S(119881)otimes119899+119898 in formula (28)

This can be written in 119898 different ways corresponding to


the composition of Δ119898minus1119894

with each of the mappings Δ119895

S(119881)otimes119899+119898minus1

rarr S(119881)otimes119899+119898 with 119894 le 119895 le 119894 + 119898 minus 1 These

are all equivalent by formula (26)

Extension to Connected GraphsWenow extend themappingsΔ119894to the vector space of all the tensors representing con-

nected graphs Blowast

= ⨁infin

120583=0B120583 where B0

= Q1 Weproceed to defineB120583 for 120583 gt 0

First given two sets 119860119899sub S(119881)

otimes119899 and 119861119901sub S(119881)

otimes119901 by119860119899119894119895119861119901

sub S(119881)otimes119899+119901minus1 with 119894 = 1 119899 119895 = 1 119901 we

denote the set of elements obtained by applying the mapping119894119895to every ordered pair (119886 isin 119860

119899 119887 isin 119861

119901) Also let B

2=

B20

and B119899= ⨁

119896max(119899)

119896=1B

119899119896 119899 ge 3 In this context for all

120583 ge 1 and 119899 ge 120583 + 1 defineBlowast

119899

120583 as follows

Blowast

119899

120583

= ⋃

1198991+sdotsdotsdot+119899

120583=119899+120583minus1

1198991

⋃

1198941=1

sdot sdot sdot

119899120583minus1

⋃

119894120583minus1=1

119899120583+sdotsdotsdot+119899

2minus120583+2

⋃

1198951=1

sdot sdot sdot

119899120583

⋃

119895120583minus1=1

B1198991

11989411198951

(B1198992

11989421198952

(sdot sdot sdot (B119899120583minus2

119894120583minus2119895120583minus2

(B119899120583minus1

119894120583minus1119895120583minus1

B119899120583

)) sdot sdot sdot)

(29)

Also define

B1015840

119899

lowast120583

= ⋃

120587isin119878119899

119878119866

120587(1)120587(119899)| 119878

119866

1119899isin B

lowast

119899

120583

(30)

where 119878119899denotes the symmetric group on the set 1 119899

and 119878119866

120587(1)120587(119899)is given by formula (16) Finally for all 120583 ge 1

define

B120583

=

infin

⨁

119899=120583+1

B1015840

119899

lowast120583

(31)

The elements of B120583 are clearly tensors representing con-nected graphs on 120583 biconnected components By (16) and(21) these may be seen as monomials on tensors represent-ing biconnected graphs with the componentwise productsdot S(119881)otimes119899 times S(119881)

otimes119899

rarr S(119881)otimes119899 so that repeated indices

correspond to cutvertices of the associated graphs In thiscontext an arbitrary connected graph say 119866 on 119899 ge 2

vertices and 120583 ge 1 biconnected components yields120583

prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886) (32)

Where for all 1 le 119886 le 120583 119861119866119886120590119886(1)120590

119886(119899119886)isin S(119881)

otimes119899 119866119886is a

biconnected graph on 2 le 119899119886le 119899 vertices represented by

119861119866119886

1119899119886

isin S(119881)otimes119899119886 and 120590

119886 1 119899

119886 rarr 119883

119886sube 1 119899 is a

bijectionWe now extend the mapping Δ = (1119899)sum

119899

119894=1Δ119894toBlowast by

requiring the mappings Δ119894to satisfy the following condition

Δ119894(

120583

prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886)) =

120583

prod

119886=1

Δ119894(119861

119866119886

120590119886(1)120590

119886(119899119886)) (33)

Given a connected graph1198661015840 the mapping Δ119894may be thought

of as a way of (a) splitting the vertex 119894 into two new vertices

numbered 119894 and 119894 + 1 and (b) distributing the biconnectedcomponents sharing the vertex 119894 between the two new ones inall the possible ways Analogously the action of themappingsΔ119898

119894consists of (a) splitting the vertex 119894 into 119898 + 1 new

vertices numbered 119894 119894 + 1 119894 + 119898 and (b) distributing thebiconnected components sharing the vertex 119894 between the119898 + 1 new ones in all the possible ways

We now combine the mappings Δ119899minus1119894

with tensors repre-senting biconnected graphs Let 119899 gt 1 119901 ge 1 and 1 le 119894 le 119901

be fixed integers Let 120587119894 1 119899 rarr 119894 119894 + 1 119894 + 119899 minus

1 119895 997891rarr 119895 + 119894 minus 1 be a bijection Let 119866 be a biconnectedgraph on 119899 vertices represented by the tensor 119861

119866

1119899isin

S(119881)otimes119899 The tensors 119861119866

119894119894+1119894+119899minus1= 119861

119866

120587119894(1)120587

119894(119899)

isin S(119881)otimes119899+119901minus1

(see formula (16)) may be viewed as operators acting onS(119881)

otimes119899+119901minus1 by multiplication In this context consider thefollowing mappings given by the composition of 119861119866

119894119894+1119894+119899minus1

with Δ119899minus1

119894

119861119866

119894119894+1119894+119899minus1∘ Δ

119899minus1

119894 S(119881)

otimes119901

997888rarr S(119881)otimes119899+119901minus1

(34)

These are the analog of the mappings 119903119866119894given in Section 3

In plain English the mappings 119861119866119894119894+1119894+119899minus1

∘ Δ119899minus1

119894produce

a connected graph with 119899 + 119901 minus 1 vertices from one with 119901

vertices in the following way

(a) split the vertex 119894 into 119899 new vertices namely 119894 119894+1 119894 + 119899 minus 1

(b) distribute the biconnected components containingthe split vertex between the 119899 new ones in all thepossible ways

(c) merge the 119899 new vertices into the graph 119866

When the graph 119866 is a 2-vertex tree the mapping 11987712

∘ Δ

coincides with the application 119871 = (120601 otimes 120601)Δ of [23] when 120601

acts on S(119881) by multiplication with a vectorTo illustrate the action of the mappings 119861119866

120587119894(1)120587

119894(119899)

∘ Δ119894

consider the graph 119867 consisting of two triangles sharing avertex Let this be represented by1198611198623

123sdot1198611198623

345isin S(119881)

otimes5 where1198623denotes a triangle represented by 1198611198623

123= 119877

12sdot 119877

23sdot 119877

13isin

S(119881)otimes3 Let 119879

2denote a 2-vertex tree represented by 119861

1198792

12=

11987712

isin S(119881)otimes2 Applying the mapping 119877

34∘ Δ

3to119867 yields

11987734

∘ Δ3(119861

1198623

123sdot 119861

1198623

345)

= 11987734

sdot Δ3(119861

1198623

123) sdot Δ

3(119861

1198623

345)

= 11987734

sdot (1198611198623

123+ 119861

1198623

124) sdot (119861

1198623

356+ 119861

1198623

456)

= 11987734

sdot 1198611198623

123sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

123sdot 119861

1198623

456

+ 11987734

sdot 1198611198623

124sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

124sdot 119861

1198623

456

(35)

Figure 7 shows the linear combination of graphs given by(35) after taking into account that the first and fourth termsas well as the second and third correspond to isomorphic


+ 2= 211987734 ∘ ( )Δ3

Figure 7The linear combination of graphs obtained by applying the mapping 11987734

∘Δ3to the cutvertex of a graph consisting of two triangles

sharing a vertex

graphs Note that 3 (resp 4) is the only cutvertex of the graphrepresented by the first (resp fourth) term while 3 and 4 areboth cutvertices of the graphs represented by the second orthird terms

7 Further Recursion Relations

Reference [7] gives two recursion formulas to generate allthe equivalence classes of trees with coefficients given bythe inverses of the orders of their groups of automorphismsOn the one hand the main formula is such that larger treesare produced from smaller ones by increasing the numberof their biconnected components by one unit On the otherhand the alternative formula is such that for all 119899 ge 2 treeson 119899 vertices are produced by connecting a vertex of a tree on 119894vertices to a vertex of a tree on 119899 minus 119894 vertices in all the possibleways Theorem 1 generalizes the main formula to connectedgraphs It is the aim of this section to derive an alternativeformula for a simplified version of the latter

Let 119866 denote a connected graph Recall the notationintroduced in Section 4 V

119866denotes the set of biconnected

components of 119866 and E(V119866) denotes the set of equivalence

classes of the graphs inV119866 Given a set119883 sub cup

119899

119901=2cup119896

119902=0119881119901119902

biconnlet 119881119899119896

119883= 119866 isin 119881

119899119896

conn E(V119866) sube E(119883) with the con-

vention 11988110

119883= (1 0) With this notation in the algebraic

setting Corollary 6 reads as follows

Theorem 7 For all 119901 gt 1 and 119902 ge 0 suppose that Φ119901119902

1119901=

sum1198661015840isin119883119901119902 120590

11986610158401198611198661015840

1119901isin B

119901119902sub S(119881)

otimes119901 with 1205901198661015840 isin Q and 119883

119901119902

sube

119881119901119902

119887119894119888119900119899119899 is such that for any equivalence classA isin E(119883119901119902

) thefollowing holds (i) there exists 1198661015840 isin A such that 120590

1198661015840 gt 0 (ii)

sum1198661015840isinA 120590

1198661015840 = 1|autA| In this context for all 119899 ge 1 and 119896 ge 0

define Ψ119899119896

isin S(119881)otimes119899 by the following recursion relation

Ψ10

= 1

Ψ1119896

= 0 if 119896 gt 0

(36)

Ψ119899119896

=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1)Φ119901119902

119894119894+1119894+119901minus1

sdotΔ119901minus1

119894(Ψ

119899minus119901+1119896minus119902

))

(37)

Then Ψ119899119896

= sum119866isin119881119899119896

119883

120572119866119878119866

1119899 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1|autC|

Now given a nonempty set 119884 sube 119881119901119902

biconn we use the fol-lowing notation

119881120583

119884= 119866 isin 119881

(119901minus1)120583+1119902120583

conn E (V119866) sube E (119884) card (V

119866) = 120583

(38)

with the convention 1198810

119884= (1 0) When we only consider

graphs whose biconnected components are all in 119884 theprevious recurrence can be transformed into one on thenumber of biconnected components

Theorem 8 Let 119901 gt 1 and 119902 ge 0 be fixed integers Let119884 sube 119881

119901119902

119887119894119888119900119899119899be a non-empty set Suppose that 120601

119901119902

1119901=

sum1198661015840isin119884

12059011986610158401198611198661015840

1119901isin B

119901119902sub S(119881)

otimes119901 with 1205901198661015840 isin Q is such

that for any equivalence class A isin E(119884) the following holds(i) there exists 1198661015840 isin A such that 120590

1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =

1|aut (A)| In this context for all 120583 ge 0 define 120595120583

isin

S(119881)otimes(119901minus1)120583+1 by the following recursion relation

1205950

= 1

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

(39)

Then 120595120583 = sum119866isin119881120583

119884

120572119866119878119866

1(119901minus1)120583+1with 120572

119866isin Q Moreover for

any equivalence classC isin E(119881120583

119884) the following holds (i) there

exists 119866 isin C such that 120572119866gt 0 (ii) sum

119866isinC 120572119866= 1|aut (C)|

Proof In this case Ψ119899119896

= 0 unless 119899 = (119901 minus 1)120583 + 1 and119896 = 119902120583 where 120583 ge 0 Therefore the recurrence of Theorem 7can be easily converted into a recurrence on the number ofbiconnected components by setting 120595120583 = Ψ

(119901minus1)120583+1119902120583

For 119901 = 2 and 119902 = 0 we recover the formula togenerate trees of [7] As in that paper and [8 23] we mayextend the result to obtain further interesting recursion rela-tions


Proposition 9 For all 120583 gt 0

120595120583

=

1

120583

times sum

1198941+sdotsdotsdot+119894119901=120583minus1

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot

(119901minus1)(1198941+sdotsdotsdot+119894119901)+119901

sum

119886119901=(119901minus1)(119894

1+sdotsdotsdot+119894119901minus1)+119901

120601119901119902

1198861119886119901

)

sdot (1205951198941

otimes sdot sdot sdot otimes 120595119894119901

)

(40)

where 119894119895= 0 120583 minus 1 for all 119895 = 1 119901

Proof Equation (40) is proved by induction on the numberof biconnected components 120583This is easily verified for 120583 = 1

1205951

= 120601119901119902

1119901sdot (1 otimes sdot sdot sdot otimes 1) = 120601

119901119902

1119901 (41)

We now assume the formula to hold for 120595120583minus1 Then formula(37) yields

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(1198941+sdotsdotsdot+119894119901minus1)+119901

120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (Δ119901minus1

119894(120595

1198941

) otimes sdot sdot sdot otimes 120595119894119901

)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941

otimes Δ119901minus1

119894(120595

1198942


)

+ sdot sdot sdot +

(119901minus1)(119894+1+sdotsdotsdot+119894119901)+119901

sum

119894=(119901minus1)(1198941+1198942+sdotsdotsdot+119894119901minus1)+119901

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941

otimes sdot sdot sdot otimes Δ119901minus1

119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot

(119901minus1)(1198941+sdotsdotsdot+119894119901)+2119901minus1

sum

119886119901=(119901minus1)(1198941+sdotsdotsdot+119894119901minus1)+2119901minus1

120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes sdot sdot sdot otimes 120595119894119901+1

)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum

1198942+sdotsdotsdot+119894119901=120583minus1minus119894

1

1198941+

120583minus1

sum

1198942=0

sum

1198941+1198943+sdotsdotsdot+119894119901=120583minus1minus119894

2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum

1198941+sdotsdotsdot+119894119901minus1=120583minus1minus119894

119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)

Note that formula (40) states that the linear combinationof all the equivalence classes of connected graphs in 119881

120583

119884is

given by summing over all the 119901-tuples of connected graphswith total number of biconnected components equal to 120583minus1gluing a vertex of each of them to distinct vertices of a graphin E(119884) in all the possible ways

8 Cayley-Type Formulas

Let 119884 sube 119881119901119902

biconn Recall that 119881120583

119884is the set of connected graphs

on 120583 biconnected components each of which has 119901 verticesand cyclomatic number 119902 and is isomorphic to a graph in 119884We proceed to use formula (40) to recover some enumerativeresults which are usually obtained via generating functionsand the Lagrange inversion formula see [3] In particularwe extend to graphs in 119881

120583

119884the result that the sum of the

inverses of the orders of the groups of automorphisms of allthe pairwise nonisomorphic trees on 119899 vertices equals 119899119899minus2119899[12 page 209]

Proposition 10 Let 119901 gt 1 and 119902 ge 0 be fixed integers Let119884 sube 119881

119901119902

119887119894119888119900119899119899be a non-empty set For all 120583 ge 0

sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)

Proof We first recall the following well-known identitiesderived from Abelrsquos binomial theorem [14]

119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1

(119899 minus 119894)119899minus119894minus1

= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1

(119910 + (119899 minus 119894))119899minus119894minus1

= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)

where 119909 and 119910 are non-zero numbers A proof may befound in [15] for instance From (45) follows that for


all 119901 gt 1 and non-zero numbers 1199091 119909

119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901

1199091sdot sdot sdot 119909

119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)

holdsTheproof proceeds by induction For119901 = 2 the identityspecializes to (45) We assume the identity (46) to hold for119901 minus 1 Then

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)

119899minus1198941minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)

We turn to the proof of formula (43) Let 119868(120583) = sumCisinE(119881120583

119884)(1

|autC|) Formula (40) induces the following recurrence for119868(120583)

119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)

sdot sdot sdot ((119901 minus 1) 119894119901+ 1) 119868 (119894

1) sdot sdot sdot 119868 (119894

119901)

(48)

We proceed to prove by induction that 119868(120583) = (((119901 minus 1)120583 +

1)120583minus2

120583)(119901sumAisinE(119884)(1|autA|))120583 The result holds for 120583 =

0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)

We assume the result to hold for all 0 le 119894 le 120583 minus 1 Then

119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1

1198941 sdot sdot sdot 119894

119901

((119901 minus 1) 1198941+ 1)

1198941minus1

sdot sdot sdot ((119901 minus 1) 119894119901+ 1)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1

sdot sdot sdot (119894119901+

1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1

times (120583 minus 1 +

119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)

where we used formula (46) in going from (52) to (53) Thiscompletes the proof of Proposition 10

The corollary is now established

Corollary 11 Let 119901 gt 1 and 119902 ge 0 be fixed integers Let 119884 sube

119881119901119902

119887119894119888119900119899119899be a non-empty set For all 120583 ge 0 119866 isin 119881

120583

119884| 119881(119866) =

1 (119901 minus 1)120583 + 1 is a set of cardinality

((119901 minus 1) 120583 + 1) ((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)

Proof The result is a straightforward application ofLagrangersquos theorem to the symmetric group on the set1 (119901 minus 1)120583 + 1 and each of its subgroups autCfor all C isin E(119881

120583

119884)

For 119901 = 2 and 119902 = 0 formula (55) specializes to Cayleyrsquosformula [11]

(120583 + 1)120583minus1

= 119899119899minus2

(56)


where 119899 = 120583 + 1 is the number of vertices of all the treeswith 120583 biconnected components In this particular case therecurrence given by formula (48) can be easily transformedinto a recurrence on the number of vertices 119899

119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)

which by (44) yields 119869(119899) = 119899119899minus2

119899 Now for 119879(119899) = 119899119869(119899)we obtain Dziobekrsquos recurrence for Cayleyrsquos formula [10 24]

119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)

Furthermore for graphs whose biconnected components areall complete graphs on 119901 vertices formula (55) yields

((119901 minus 1) 120583 + 1)((119901 minus 1) 120583 + 1)120583minus2

(119901 minus 1) 120583

120583

(59)

in agreement with Husimirsquos result for this particular case [25](see also [26]) Also when the biconnected components areall cycles of length 119901 we recover a particular case of Lerouxrsquosresult [27]


2120583120583

(60)

Acknowledgment

The research was supported through the fellowship SFRHBPD482232008 provided by the Portuguese Foundation forScience and Technology (FCT)

References

[1] R C Read ldquoA survey of graph generation techniquesrdquo in Com-binatorial Mathematics 8 vol 884 of Lecture Notes in Math pp77ndash89 Springer Berlin Germany 1981

[2] F Harary and E M Palmer Graphical Enumeration AcademicPress New York NY USA 1973

[3] F Bergeron G Labelle and P Leroux Combinatorial Speciesand Tree-Like Structures vol 67 Cambridge University PressCambridge UK 1998

[4] C Itzykson and J B Zuber Quantum Field Theory McGraw-Hill New York NY USA 1980

[5] C Jordan ldquoSur les assemblages de lignesrdquo Journal fur die Reineund Angewandte Mathematik vol 70 pp 185ndash190 1869

[6] R C Read ldquoEvery one a winner or how to avoid isomorphismsearch when cataloguing combinatorial configurations Algo-rithmic aspects of combinatorics (Conference in VancouverIsland BC Canada 1976)rdquo Annals of Discrete Mathematics vol2 pp 107ndash120 1978

[7] A Mestre and R Oeckl ldquoCombinatorics of 119899-point functionsvia Hopf algebra in quantum field theoryrdquo Journal ofMathemat-ical Physics vol 47 no 5 p 052301 16 2006

[8] A Mestre and R Oeckl ldquoGenerating loop graphs via Hopfalgebra in quantum field theoryrdquo Journal of MathematicalPhysics vol 47 no 12 Article ID 122302 p 14 2006

[9] A Mestre ldquoCombinatorics of 1-particle irreducible n-pointfunctions via coalgebra in quantum field theoryrdquo Journal ofMathematical Physics vol 51 no 8 Article ID 082302 2010

[10] O Dziobek ldquoEine formel der substitutionstheorierdquo Sitzungs-berichte der Berliner Mathematischen Gesellschaft vol 17 pp64ndash67 1947

[11] A Cayley ldquoA theorem on treesrdquo Quarterly Journal of Pure andApplied Mathematics vol 23 pp 376ndash378 1889

[12] G Polya ldquoKombinatorische Anzahlbestimmungen fur Grup-pen Graphen und chemische Verbindungenrdquo Acta Mathemat-ica vol 68 pp 145ndash254 1937

[13] R Tarjan ldquoDepth-first search and linear graph algorithmsrdquoSIAM Journal on Computing vol 1 no 2 pp 146ndash160 1972

[14] N Abel ldquoBeweis eines Ausdruckes von welchem die Binomial-Formel ein einzelner Fall istrdquo Journal fur die Reine und Ange-wandte Mathematik vol 1 pp 159ndash160 1826

[15] L Szekely ldquoAbelrsquos binomial theoremrdquo httpwwwmathscedusimszekelyabelpdf

[16] R Diestel Graph Theory vol 173 Springer Berlin Germany5rd edition 2005

[17] M J Atallah and S Fox Algorithms andTheory of ComputationHandbook CRC Press Boca Raton Fla USA 1998

[18] G Kirchhoff ldquoUber die Auflosung der Gleichungen aufwelche man bei der Untersuchung der linearen Vertheilunggalvanischer Strome gefuhrt wirdrdquo Annual Review of PhysicalChemistry vol 72 pp 497ndash508 1847

[19] A Mestre ldquoGenerating connected and 2-edge connectedgraphsrdquo Journal of Graph Algorithms and Applications vol 13no 2 pp 251ndash281 2009

[20] J Cuntz R Meyer and J M Rosenberg Topological andBivariantK-Theory vol 36 Birkhauser Basel Switzerland 2007

[21] C Kassel Quantum Groups vol 155 Springer New York NYUSA 1995

[22] J-L Loday Cyclic Homology vol 301 Springer Berlin Ger-many 1992

[23] M Livernet ldquoA rigidity theorem for pre-Lie algebrasrdquo Journal ofPure and Applied Algebra vol 207 no 1 pp 1ndash18 2006

[24] J W Moon ldquoVarious proofs of Cayleyrsquos formula for countingtreesrdquo in A Seminar on GraphTheory pp 70ndash78 Holt Rinehartamp Winston New York NY USA 1967

[25] K Husimi ldquoNote on Mayersrsquo theory of cluster integralsrdquo TheJournal of Chemical Physics vol 18 pp 682ndash684 1950

[26] J Mayer ldquoEquilibrium statistical mechanicsrdquo in The Interna-tional Encyclopedia of Physical Chemistry and Chemical PhysicsPergamon Press Oxford UK 1968

[27] P Leroux ldquoEnumerative problems inspired by Mayerrsquos theoryof cluster integralsrdquo Electronic Journal of Combinatorics vol 11no 1 research paper 32 p 28 2004

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


representative is a graph on 119898 edges say 119866 we show thatevery one of the119898 edges of the graph119866 adds 1(119898sdot |aut119866|) tothe coefficient of 119866 To this end we use the fact that labeledvertices are held fixed under any automorphism

Moreover in the algebraic representation framework theresult yields a recurrence to generate linear combinations oftensors over the rational numbers Each tensor represents aconnected graph As required these linear combinations havethe property that the sum of the coefficients of all the tensorsrepresenting isomorphic graphs is the inverse of the orderof their group of automorphisms In this context tensorsrepresenting generated graphs are factorized into tensorsrepresenting their biconnected components As in [7ndash9] akey feature of this result is its close relation to the algorithmicdescription of the computations involved Indeed it is easyto read off from this scheme not only algorithms to performthe computations but even data structures relevant for animplementation

Furthermore we prove that when we only considerthe restricted class of connected graphs whose biconnectedcomponents all have say 119901 vertices and 119903 edges the cor-responding recurrence has an alternative expression whichrelates connected graphs on 120583 biconnected components withall the 119901-tuples of their connected subgraphs with totalnumber of biconnected components equal to 120583 minus 1 Thisinduces a recurrence for the sum of the inverses of the ordersof the groups of automorphisms of all the equivalence classesof connected graphs on 120583 biconnected components with thatpropertyThe proof uses an identity related to Abelrsquos binomialtheorem [14 15] and generalizes Dziobekrsquos induction proof ofCayleyrsquos formula [10]

This paper is organized as follows Section 2 reviews thebasic concepts of graph theory underlyingmuch of the paperSection 3 contains the definitions of the elementary graphtransformations to be used in the following Section 4 gives arecursion formula for generating all the equivalence classes ofconnected graphs in terms of their biconnected componentsSections 5 and 6 review the algebraic representation andsome of the linear mappings introduced in [7 9] Section 7derives an algebraic expression for the recurrence of Section4 and for the particular case in which graphs are such thattheir biconnected components are all graphs on the samevertex and edge numbers An alternative formulation for thelatter is also given Finally Section 8 proves a Cayley-typeformula for graphs of that kind

2 Basics

We briefly review the basic concepts of graph theory that arerelevant for the following sectionsMore detailsmay be foundin any standard textbook on graph theory such as [16]

Let 119860 and 119861 denote sets By [119860]2 we denote the set of allthe 2-element subsets of 119860 Also by 2119860 we denote the powerset of 119860 that is the set of all the subsets of 119860 By card119860 wedenote the cardinality of the set 119860 Furthermore we recallthat the symmetric difference of the sets 119860 and 119861 is given by119860Δ119861 = (119860 cup 119861) (119860 cap 119861)

Here a graph is a pair 119866 = (119881 119864) where 119881 sub N is afinite set and 119864 sube [119881]

2 Thus the elements of 119864 are 2-element

subsets of 119881 The elements of 119881 and 119864 are called verticesand edges respectively In the following the vertex set of agraph 119866 will often be referred to as 119881(119866) the edge set as119864(119866) The cardinality of119881(119866) is called the order of 119866 writtenas |119866| A vertex 119907 is said to be incident with an edge 119890 if119907 isin 119890 Then 119890 is an edge at 119907 The two vertices incidentwith an edge are its endvertices Moreover the degree of avertex 119907 is the number of edges at 119907 Two vertices 119907 and 119906

are said to be adjacent if 119907 119906 isin 119864 If all the vertices of 119866are pairwise adjacent then 119866 is said to be complete A graph119866lowast is called a subgraph of a graph 119866 if 119881(119866lowast) sube 119881(119866) and

119864(119866lowast

) sube 119864(119866) A path is a graph 119875 on 119899 ge 2 vertices suchthat 119864(119875) = 119907

1 1199072 119907

2 1199073 119907

119899minus1 119907119899 119907

119895isin 119881(119875) for

all 119895 = 1 119899 The vertices 1199071and 119907

119899have degree 1 while

the vertices 1199072 119907

119899minus1have degree 2 In this context the

vertices 1199071and 119907

119899are linked by 119875 and called the endpoint

verticesThe vertices 1199072 119907

119899minus1are called the inner vertices

A cycle is a graph 119862 on 119899 gt 2 vertices such that 119864(119862) =

1199071 1199072 119907

2 1199073 119907

119899minus1 119907119899 119907

119899 1199071 119907

119895isin 119881(119862) for all

119895 = 1 119899 every vertex having degree 2 A graph issaid to be connected if every pair of vertices is linked bya path Otherwise it is disconnected Given a graph 119866 amaximal connected subgraph of 119866 is called a component of119866 Furthermore given a connected graph a vertex whoseremoval (together with its incident edges) disconnects thegraph is called a cutvertex A graph that remains connectedafter erasing any vertex (together with incident edges) (respany edge) is said to be 2-connected (resp 2-edge connected)A 2-connected graph (resp 2-edge connected graph) is alsocalled biconnected (resp edge-biconnected) Given a con-nected graph119867 a biconnected component of119867 is a maximalsubset of edges such that the induced subgraph is biconnected(see [17 Section 64] for instance) Here we consider that anisolated vertex is by convention a biconnected graphwith nobiconnected components

Moreover given a graph 119866 the set 2119864(119866) is a vector spaceover the field Z

2such that vector addition is given by the

symmetric difference The cycle space C(119866) of the graph 119866

is defined as the subspace of 2119864(119866) generated by all the cyclesin 119866 The dimension ofC(119866) is called the cyclomatic numberof the graph119866 We recall that dimC(119866) = card119864(119866)minus |119866|+ 119888where 119888 denotes the number of connected components of thegraph 119866 [18]

We now introduce a definition of labeled graph Let 119871be a finite set Here a labeling of a graph 119866 is a mapping119897 119881(119866) rarr 2

119871 such that cup119907isin119881(119866)

119897(119907) = 119871 and 119897(119907) cap 119897(1199071015840

) = 0

for all 119907 1199071015840 isin 119881(119866) with 119907 = 1199071015840 In this context 119871 is called

a label set while the graph 119866 is said to be labeled with 119871 orsimply a labeled graph In the sequel a labeling of a graph 119866

will be referred to as 119897119866 Moreover an unlabeled graph is one

labeled with the empty setFurthermore an isomorphism between two graphs 119866 and

119866lowast is a bijection 120593 119881(119866) rarr 119881(119866

lowast

) which satisfies the fol-lowing conditions

(i) 119907 1199071015840 isin 119864(119866) if and only if 120593(119907) 120593(1199071015840) isin 119864(119866lowast

)

(ii) 119871 cap 119897119866(119907) = 119871 cap 119897

119866lowast(120593(119907))






119881119899119896119871

= 119866 |119866| = 119899 dimC (119866) = 119896 119866 is labeled

by 119897119866 119881 (119866) 997888rarr 2

119871

(1)


119899119896119871


119899119896119871



119899119896119871 and the other on 1198811199011199021198711015840

with 119899 = 119901 or 119896 = 119902 or119871 = 119871



119899119896119871

conn Let119881(119866) = 119907

119894119894=1119899





joint sets L = 119911119894= (K

(1)

119894 K

(119901)

119894) cup

119901

119897=1K

(119897)

119894

= K119894and K

(119897)

119894cap K

(1198971015840

)

119894= 0 forall119897 119897

1015840

= 1 119901 with 119897 = 119897

1015840


119866(119907119894) into 119901

disjoint sets J = 119908119894= (119897

119866(119907119894)(1)

119897119866(119907119894)(119901)

)

cup119901

119897=1119897119866(119907119894)(119897)

= 119897119866(119907119894) and 119897

119866(119907119894)(119897)

cap 119897119866(119907119894)(1198971015840

)

= 0 forall119897

1198971015840

= 1 119901 with 119897 = 1198971015840




119897119897=1119901


119903

119866

119894 Q119881

119899119896119871

conn 997888rarr Q119881119899+119901minus1119896+119902119871

conn

119866 997891997888rarr sum

119911119894isinL119908

119894isinJ

119866119911119894

119908119894

(2)

where the graphs 119866119911119894119908119894


(a) 119881(119866119911119894119908119894

) = 119881(119866) 119907119894 cup 119881(119866)

(b) 119864(119866119911119894119908119894

) = 119864(119866) cup 119909 119910 isin 119864(119866) 119907119894notin 119909 119910

119901

⋃

119897=1

119909 119906119897 119909 119907

119894 isin 119864 (119866) 119909 isin cup


119894

119881 (119867lowast

) (3)

(c) 119897119866119911119894

119908119894

|119881(119866)119907

119894= 119897

119866|119881(119866)119907

119894and 119897

119866119911119894

119908119894

(119906119897) = 119897

119866(119907119894)(119897)

for all 119897 = 1 119901


119899119896119871







119866isin119883120572119866119866 where 120572

119866isin Q

we define

119903120599

119894= sum

119866isin119883

120572119866119903119866

119894 (4)



119899119896119871


1199011199021198711015840


119907isin119881(119866)119897119866(119907) Define

119888119866 119881

119899119896119871


conn 119866 997891997888rarr 119866lowast

(5)



isin

N 1199071015840


cup 119909 119907 119909 119910 isin 119864 (119866) 119864 (119866) and 119910 isin 119881 (119866) (6)

(c) 119897119866lowast |119881(119866)119881(

119866)

= 119897119866|119881(119866)119881(

119866)

and 119897119866lowast(119907) =

cup119906isin119881(

119866)119897119866(119906)


1198624


119899119896119871 Let119881(119866) = 119907

119894119894=1119899


1015840


(1198711015840(1)

1198711015840(119899)

) cup119899

119895=11198711015840(119895)

= 1198711015840 and 119871

1015840(119894)

cap 1198711015840(119895)

=


1205851198711015840 Q119881

119899119896119871

997888rarr Q119881119899119896119871cup119871

1015840

119866 997891997888rarr sum

119910isinI

119866119910 (7)


119903cutvertex

= 4 + 8 + 4)(



=)(119888




(a) 119881(119866119910) = 119881(119866)

(b) 119864(119866119910) = 119864(119866)

(c) 119897119866119910

(119907119894) = 119897

119866(119907119894) cup 119871

1015840(119894) for all 119894 = 1 119899 and119907119894isin 119881(119866)


119899119896119871 by line-arity







=

sum1198661015840isin119881119901119902

119887119894119888119900119899119899

1205901198661015840119866


classA isin E(119881119901119902



1198661015840isinA 120590



isin

Q119881119899119896119871


12057310119871

119888119900119899119899= 119866 119908ℎ119890119903119890 119866 = (1 0) 119897

119866(1) = 119871

1205731119896119871

119888119900119899119899= 0 119894119891 119896 gt 0

120573119899119896119871

119888119900119899119899=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1) 119903

120573119901119902

119887119894119888119900119899119899

119894(120573

119899minus119901+1119896minus119902119871

119888119900119899119899))

(8)

Then 120573119899119896119871119888119900119899119899

= sum119866isin119881119899119896119871

119888119900119899119899

120572119866119866 with 120572




119866gt 0 (ii) sum

119866isinC 120572119866= 1|autC|



119866isin119881119899119896119871



119866gt 0




biconn1205901198661015840119866

1015840 with


119901119902

biconn119894

read as

119903

120573119901119902

biconn119894

= sum

1198661015840isin119881119901119902

biconn

12059011986610158401199031198661015840

119894 (9)


120573119901119902

biconn119894


isin 119881119899minus119901+1119896minus119902119871



119866lowast is the



119907isin119881(119866)119897119866(119907)



notin


119899minus119901+1119896minus119902119871






cong

119888119866(119866) and 120584

119867lowast gt 0 Let 119907

119895isin 119881(119867

lowast




isin 119881119901119902




119867gt 0


119888119900119899119899= sum

119866isin119881119899119896119871

119888119900119899119899

120572119866119866 be




sum119866isinC 120572

119866= 1




119866gt 0 where 120572


in 120573119899119896119871







1199011199021198711015840


= cup119907isin119881(

119866)119897119866(119907) Let 1198981015840

= card119864(119866) so that 1198981015840

=


1015840


119899minus119901+1119896minus119902119871


119888119866(119866)



conn120584119866lowast119866


that 119867 cong 119888119866(119866) and 120584



119866lowast = 1 Now let 119907



119901119902


119901119902


1015840




1015840 with

119897119888119866(119866)


1198661015840

119895(119867)



1198661015840



= 120584119867


1198661015840



119899minus119901+1119896minus119902119871


sum

119866isinC

120572lowast


119866lowastisinD

120584119866lowast

= |autA|

(10)


119866isinC 120572119866is 1198981015840



119866isinC 120572119866





= 0 Then 120573119899119896119871cup1198711015840

119888119900119899119899= 120585

1198711015840(120573

119899119896119871

119888119900119899119899)

Proof Let 1205851198711015840 119881

119899119896119871

rarr 119881119899119896119871cup119871

1015840

and 1205851198711015840 119881

119899minus119901+1119896minus119902119871

rarr

119881119899minus119901+1119896minus119902119871cup119871

1015840


1198711015840 ∘ 119903

120573119901119902

biconn119894

= 119903

120573119901119902

biconn119894

∘1205851198711015840


119888119900119899119899= sum

119866isin119881119899119896119871

119888119900119899119899


(8) Let C isin E(119881119899119896119871119888119900119899119899


119866= 1|aut (C)|



119866(119881

1015840


1015840



such that 119897|119881(119866)119881

1015840 = 119897119866|119881(119866)119881

1015840 and 119897(119907) = 0 for all 119907 isin 1198811015840


isin 119881119899119896119871cup119871

1015840


1015840


1015840

conn = sum1198661015840isin119881119899119896119871cup119871

1015840

conn1205841198661015840119866


3 sum1198661015840isinD 120584



119871cup1198711015840

119895 = 1 119879 such that119897119895|119881(119866)119881

1015840 = 119897119866|119881(119866)119881

1015840 119897119895(119907) = 0 for all 119907 isin 119881

1015840 and 119866119895isin D


119895 Clearly

120584119866119895

= 120572119866gt 0 where 120584

119866119895


119899119896119871cup1198711015840

conn Define 119891

119895 119866 997891rarr 119866



12057310conn =

12057320conn =

12057330conn =

12057340conn =

12057341conn =

12057331conn =

12057340conn =

1

2

1

3

1

8

+1

2

+1

2

+1

4

+1

3

1

2

1

2

1

2




sum

1198661015840isinD

1205841198661015840 =

119879

sum

119895=1

sum

119866isinC

120584119891119895(119866)

= 119879sum

119866isinC

120572119866= 1 (11)



sum119866isinC 120572

119866= 1|aut (C)|





119901=2cup119896

119902=0119881119901119902

biconn let119881119899119896

119883=

119866 isin 119881119899119896


10

119883=



=

sum1198661015840isin119883119901119902 120590

1198661015840119866

1015840 with 1205901198661015840 isin Q and 119883

119901119902

sube 119881119901119902

119887119894119888119900119899119899 is such that



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin

Q119881119899119896


12057410

119888119900119899119899= 119866 119908ℎ119890119903119890 119866 = (1 0)

1205741119896

119888119900119899119899= 0 if 119896 gt 0

120574119899119896

119888119900119899119899=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1) 119903

120574119901119902

119887119894119888119900119899119899

119894(120574

119899minus119901+1119896minus119902

conn ))

(12)

Then 120574119899119896119888119900119899119899

= sum119866isin119881119899119896

119883

120572119866119866 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1| autC|







infin

119896=0S119896(119881) where





997888rarr S(119881)otimes119899


119899 1199041015840


1015840

119899)

997891997888rarr 11990411199041015840


1198991199041015840

119899

(13)




1

(a)

1 2

(b)

3

21

(c)


12isin S(119881)


12sdot 119877

13sdot 119877

23isin S(119881)

otimes3


otimes119899

119877119894119895= 1

otimes119894minus1



(14)


119894119895isin S(119881)

otimes119899 and 119877119894119895

isin S(119881)otimes1198991015840

with 119899 = 1198991015840



(ii) a tensor 119877119894119895


a graph on 1 119899


and 119864(119866) = 119894119896 119895119896119896=1119898



S(119881)otimes119899



119894119896119895119896



1119899isin S(119881)


119894119896119895119896

119878119866

1119899=

119898

prod

119896=1

119877119894119896119895119896

(15)


1015840



1015840

119878119866

120590(1)120590(119899)=

119898

prod

119896=1

119877120590(119894119896)120590(119895119896) (16)



1

3 4

2

(a)

5

3

4

1

2

(b)






5 3 997891rarr 4 4 997891rarr 2


1015840



119896=0S(119881)





119899) ∙ (119904

1015840


1015840

1198991015840)


119899otimes 119904

1015840


1015840

1198991015840

(18)

where 119904119894 1199041015840




1otimes 119904

2997891rarr 119904

2otimes 119904

1


otimes119896minus1



1015840 we define ∙119894119895


rarr S(119881)otimes119899+1198991015840



119899) ∙119894119895(1199041015840


1015840

1198991015840)

= (120591119899minus1

∘ sdot sdot sdot ∘ 120591119894) (119904


119899)


119895minus1) (119904

1015840


1015840

1198991015840)



119899otimes 119904

119894

otimes 1199041015840

119895otimes 119904

1015840


1199041015840


1015840

1198991015840

(19)

where 119904119894(resp 1199041015840

119895) means that 119904

119894(resp 1199041015840



isin S(119881)otimes119899 and 119878119866

1015840

11198991015840 isin

S(119881)otimes1198991015840


119878119866

1119899∙1198941198951198781198661015840

11198991015840 = 119878

119866

120590(1)120590(119899)sdot 1198781198661015840

1205901015840(1)120590

1015840(1198991015840) (20)

where 119878119866120590(1)120590(119899)

1198781198661015840

1205901015840(1)120590

1015840(1198991015840)isin S(119881)

otimes119899+1198991015840

120590(119896) = 119896 if 1 le 119896 lt

119894 120590(119894) = 119899 120590(119896) = 119896 minus 1 if 119894 lt 119896 le 119899 and 1205901015840

(119896) = 119896 + 119899 + 1

if 1 le 119896 lt 119895 1205901015840(119895) = 119899 + 1 1205901015840(119896) = 119896 + 119899 if 119895 lt 119896 le 1198991015840


1

3 41

32 2119866prime119866

(a)

5

6

1

4 3

2

119867

(b)



and 1198781198661015840

123isin S(119881)


=

119878119866

1234321198781198661015840

123isin S(119881)

otimes6


∙1198941198951198781198661015840



otimes119894minus1


S(119881)otimes119899




119894119895= sdot

119899∘ ∙

119894119895 S(119881)

otimes119899

times S(119881)otimes1198991015840

997888rarr S(119881)otimes119899+1198991015840

minus1

(21)

The tensor 1198781198661119899

1198941198951198781198661015840


1015840

minus 1


6 Linear Mappings


119899119896sub S(119881)


119899119896

biconn Let B =

B10⨁B

20⨁

infin

119899=3⨁

119896max(119899)

119896=1B

119899119896sub T(S(119881)) where 119896max(119899) =



Δ (1) = 1 otimes 1

Δ (119861119866

1119899) =

1

119899

119899

sum

119894=1

Δ119894(119861

119866

1119899) if 119899 gt 1

(22)


119866

1119899isin S(119881)



(i) 120590119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894

119895 + 1 if 119894 + 1 le 119895 le 119899

(ii) 120584119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894 minus 1

119895 + 1 if 119894 le 119895 le 119899

(23)


otimes119899


equation

Δ119894(119861

119866

1119899) = 119861

119866

120590119894(1)120590

119894(119899)

+ 119861119866

120584119894(1)120584

119894(119899)

= 119861119866

1119894+1119894+2119899+1+ 119861

119866

1119894119894+1119899+1

(24)


1119894119899+1(resp

119861119866

1119894+1119894+2119899+1





isin S(119881)otimes1198991015840


1015840


Δ119894(119861

119866

120587(1)120587(119899)) = 119861

119866

120584119894(120587(1))120584

119894(120587(119899))

(25)



Δ119894∘ Δ

119894= Δ

119894+1∘ Δ

119894 (26)


S(119881)otimes119899

rarr


and Δ119894 S(119881)

otimes119899+1



S(119881)otimes119899+1



119894 Δ119898

119894

S(119881)otimes119899


Δ1

119894= Δ

119894 (27)

Δ119898

119894= Δ

119894∘ Δ

119898minus1

119894 (28)

where Δ119894

S(119881)otimes119899+119898minus1






S(119881)otimes119899+119898minus1





= ⨁infin

120583=0B120583 where B0



otimes119899 and 119861119901sub S(119881)

otimes119901 by119860119899119894119895119861119901



119899 119887 isin 119861

119901) Also let B

2=

B20

and B119899= ⨁

119896max(119899)

119896=1B



119899

120583 as follows

Blowast

119899

120583

= ⋃


120583=119899+120583minus1

1198991

⋃

1198941=1

sdot sdot sdot

119899120583minus1

⋃

119894120583minus1=1

119899120583+sdotsdotsdot+119899

2minus120583+2

⋃

1198951=1

sdot sdot sdot

119899120583

⋃

119895120583minus1=1

B1198991

11989411198951

(B1198992

11989421198952


119894120583minus2119895120583minus2

(B119899120583minus1

119894120583minus1119895120583minus1

B119899120583

)) sdot sdot sdot)

(29)

Also define

B1015840

119899

lowast120583

= ⋃

120587isin119878119899

119878119866

120587(1)120587(119899)| 119878

119866

1119899isin B

lowast

119899

120583

(30)


and 119878119866


define

B120583

=

infin

⨁

119899=120583+1

B1015840

119899

lowast120583

(31)


otimes119899




prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886) (32)


119886(119899119886)isin S(119881)

otimes119899 119866119886is a


119861119866119886

1119899119886

isin S(119881)otimes119899119886 and 120590

119886 1 119899

119886 rarr 119883

119886sube 1 119899 is a


119899



Δ119894(

120583

prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886)) =

120583

prod

119886=1

Δ119894(119861

119866119886

120590119886(1)120590

119886(119899119886)) (33)










119866

1119899isin


119894119894+1119894+119899minus1= 119861

119866

120587119894(1)120587

119894(119899)




119894119894+1119894+119899minus1

with Δ119899minus1

119894

119861119866

119894119894+1119894+119899minus1∘ Δ

119899minus1

119894 S(119881)

otimes119901


(34)



∘ Δ119899minus1

119894produce







∘ Δ



120587119894(1)120587

119894(119899)

∘ Δ119894


123sdot1198611198623

345isin S(119881)


123= 119877

12sdot 119877

23sdot 119877

13isin

S(119881)otimes3 Let 119879


1198792

12=

11987712


34∘ Δ

3to119867 yields

11987734

∘ Δ3(119861

1198623

123sdot 119861

1198623

345)

= 11987734

sdot Δ3(119861

1198623

123) sdot Δ

3(119861

1198623

345)

= 11987734

sdot (1198611198623

123+ 119861

1198623

124) sdot (119861

1198623

356+ 119861

1198623

456)

= 11987734

sdot 1198611198623

123sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

123sdot 119861

1198623

456

+ 11987734

sdot 1198611198623

124sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

124sdot 119861

1198623

456

(35)



+ 2= 211987734 ∘ ( )Δ3



sharing a vertex








119899

119901=2cup119896

119902=0119881119901119902

biconnlet 119881119899119896

119883= 119866 isin 119881

119899119896


vention 11988110




1119901=

sum1198661015840isin119883119901119902 120590

11986610158401198611198661015840

1119901isin B

119901119902sub S(119881)


119901119902

sube

119881119901119902



1198661015840 gt 0 (ii)

sum1198661015840isinA 120590


define Ψ119899119896


Ψ10

= 1

Ψ1119896

= 0 if 119896 gt 0

(36)

Ψ119899119896

=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1)Φ119901119902

119894119894+1119894+119901minus1

sdotΔ119901minus1

119894(Ψ

119899minus119901+1119896minus119902

))

(37)

Then Ψ119899119896

= sum119866isin119881119899119896

119883

120572119866119878119866

1119899 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1|autC|



119881120583

119884= 119866 isin 119881

(119901minus1)120583+1119902120583


119866) = 120583

(38)





119901119902


119901119902

1119901=

sum1198661015840isin119884

12059011986610158401198611198661015840

1119901isin B

119901119902sub S(119881)



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin


1205950

= 1

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

(39)

Then 120595120583 = sum119866isin119881120583

119884

120572119866119878119866

1(119901minus1)120583+1with 120572





119866isinC 120572119866= 1|aut (C)|



(119901minus1)120583+1119902120583




120595120583

=

1

120583

times sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(119894


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(40)



1205951

= 120601119901119902


119901119902

1119901 (41)


120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


119894(120595

1198941


)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941


119894(120595

1198942


)

+ sdot sdot sdot +


sum


Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119881119899119896119871

= 119866 |119866| = 119899 dimC (119866) = 119896 119866 is labeled

by 119897119866 119881 (119866) 997888rarr 2

119871

(1)


119899119896119871


119899119896119871



119899119896119871 and the other on 1198811199011199021198711015840

with 119899 = 119901 or 119896 = 119902 or119871 = 119871



119899119896119871

conn Let119881(119866) = 119907

119894119894=1119899





joint sets L = 119911119894= (K

(1)

119894 K

(119901)

119894) cup

119901

119897=1K

(119897)

119894

= K119894and K

(119897)

119894cap K

(1198971015840

)

119894= 0 forall119897 119897

1015840

= 1 119901 with 119897 = 119897

1015840


119866(119907119894) into 119901

disjoint sets J = 119908119894= (119897

119866(119907119894)(1)

119897119866(119907119894)(119901)

)

cup119901

119897=1119897119866(119907119894)(119897)

= 119897119866(119907119894) and 119897

119866(119907119894)(119897)

cap 119897119866(119907119894)(1198971015840

)

= 0 forall119897

1198971015840

= 1 119901 with 119897 = 1198971015840




119897119897=1119901


119903

119866

119894 Q119881

119899119896119871

conn 997888rarr Q119881119899+119901minus1119896+119902119871

conn

119866 997891997888rarr sum

119911119894isinL119908

119894isinJ

119866119911119894

119908119894

(2)

where the graphs 119866119911119894119908119894


(a) 119881(119866119911119894119908119894

) = 119881(119866) 119907119894 cup 119881(119866)

(b) 119864(119866119911119894119908119894

) = 119864(119866) cup 119909 119910 isin 119864(119866) 119907119894notin 119909 119910

119901

⋃

119897=1

119909 119906119897 119909 119907

119894 isin 119864 (119866) 119909 isin cup


119894

119881 (119867lowast

) (3)

(c) 119897119866119911119894

119908119894

|119881(119866)119907

119894= 119897

119866|119881(119866)119907

119894and 119897

119866119911119894

119908119894

(119906119897) = 119897

119866(119907119894)(119897)

for all 119897 = 1 119901


119899119896119871







119866isin119883120572119866119866 where 120572

119866isin Q

we define

119903120599

119894= sum

119866isin119883

120572119866119903119866

119894 (4)



119899119896119871


1199011199021198711015840


119907isin119881(119866)119897119866(119907) Define

119888119866 119881

119899119896119871


conn 119866 997891997888rarr 119866lowast

(5)



isin

N 1199071015840


cup 119909 119907 119909 119910 isin 119864 (119866) 119864 (119866) and 119910 isin 119881 (119866) (6)

(c) 119897119866lowast |119881(119866)119881(

119866)

= 119897119866|119881(119866)119881(

119866)

and 119897119866lowast(119907) =

cup119906isin119881(

119866)119897119866(119906)


1198624


119899119896119871 Let119881(119866) = 119907

119894119894=1119899


1015840


(1198711015840(1)

1198711015840(119899)

) cup119899

119895=11198711015840(119895)

= 1198711015840 and 119871

1015840(119894)

cap 1198711015840(119895)

=


1205851198711015840 Q119881

119899119896119871

997888rarr Q119881119899119896119871cup119871

1015840

119866 997891997888rarr sum

119910isinI

119866119910 (7)


119903cutvertex

= 4 + 8 + 4)(



=)(119888




(a) 119881(119866119910) = 119881(119866)

(b) 119864(119866119910) = 119864(119866)

(c) 119897119866119910

(119907119894) = 119897

119866(119907119894) cup 119871

1015840(119894) for all 119894 = 1 119899 and119907119894isin 119881(119866)


119899119896119871 by line-arity







=

sum1198661015840isin119881119901119902

119887119894119888119900119899119899

1205901198661015840119866


classA isin E(119881119901119902



1198661015840isinA 120590



isin

Q119881119899119896119871


12057310119871

119888119900119899119899= 119866 119908ℎ119890119903119890 119866 = (1 0) 119897

119866(1) = 119871

1205731119896119871

119888119900119899119899= 0 119894119891 119896 gt 0

120573119899119896119871

119888119900119899119899=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1) 119903

120573119901119902

119887119894119888119900119899119899

119894(120573

119899minus119901+1119896minus119902119871

119888119900119899119899))

(8)

Then 120573119899119896119871119888119900119899119899

= sum119866isin119881119899119896119871

119888119900119899119899

120572119866119866 with 120572




119866gt 0 (ii) sum

119866isinC 120572119866= 1|autC|



119866isin119881119899119896119871



119866gt 0




biconn1205901198661015840119866

1015840 with


119901119902

biconn119894

read as

119903

120573119901119902

biconn119894

= sum

1198661015840isin119881119901119902

biconn

12059011986610158401199031198661015840

119894 (9)


120573119901119902

biconn119894


isin 119881119899minus119901+1119896minus119902119871



119866lowast is the



119907isin119881(119866)119897119866(119907)



notin


119899minus119901+1119896minus119902119871






cong

119888119866(119866) and 120584

119867lowast gt 0 Let 119907

119895isin 119881(119867

lowast




isin 119881119901119902




119867gt 0


119888119900119899119899= sum

119866isin119881119899119896119871

119888119900119899119899

120572119866119866 be




sum119866isinC 120572

119866= 1




119866gt 0 where 120572


in 120573119899119896119871







1199011199021198711015840


= cup119907isin119881(

119866)119897119866(119907) Let 1198981015840

= card119864(119866) so that 1198981015840

=


1015840


119899minus119901+1119896minus119902119871


119888119866(119866)



conn120584119866lowast119866


that 119867 cong 119888119866(119866) and 120584



119866lowast = 1 Now let 119907



119901119902


119901119902


1015840




1015840 with

119897119888119866(119866)


1198661015840

119895(119867)



1198661015840



= 120584119867


1198661015840



119899minus119901+1119896minus119902119871


sum

119866isinC

120572lowast


119866lowastisinD

120584119866lowast

= |autA|

(10)


119866isinC 120572119866is 1198981015840



119866isinC 120572119866





= 0 Then 120573119899119896119871cup1198711015840

119888119900119899119899= 120585

1198711015840(120573

119899119896119871

119888119900119899119899)

Proof Let 1205851198711015840 119881

119899119896119871

rarr 119881119899119896119871cup119871

1015840

and 1205851198711015840 119881

119899minus119901+1119896minus119902119871

rarr

119881119899minus119901+1119896minus119902119871cup119871

1015840


1198711015840 ∘ 119903

120573119901119902

biconn119894

= 119903

120573119901119902

biconn119894

∘1205851198711015840


119888119900119899119899= sum

119866isin119881119899119896119871

119888119900119899119899


(8) Let C isin E(119881119899119896119871119888119900119899119899


119866= 1|aut (C)|



119866(119881

1015840


1015840



such that 119897|119881(119866)119881

1015840 = 119897119866|119881(119866)119881

1015840 and 119897(119907) = 0 for all 119907 isin 1198811015840


isin 119881119899119896119871cup119871

1015840


1015840


1015840

conn = sum1198661015840isin119881119899119896119871cup119871

1015840

conn1205841198661015840119866


3 sum1198661015840isinD 120584



119871cup1198711015840

119895 = 1 119879 such that119897119895|119881(119866)119881

1015840 = 119897119866|119881(119866)119881

1015840 119897119895(119907) = 0 for all 119907 isin 119881

1015840 and 119866119895isin D


119895 Clearly

120584119866119895

= 120572119866gt 0 where 120584

119866119895


119899119896119871cup1198711015840

conn Define 119891

119895 119866 997891rarr 119866



12057310conn =

12057320conn =

12057330conn =

12057340conn =

12057341conn =

12057331conn =

12057340conn =

1

2

1

3

1

8

+1

2

+1

2

+1

4

+1

3

1

2

1

2

1

2




sum

1198661015840isinD

1205841198661015840 =

119879

sum

119895=1

sum

119866isinC

120584119891119895(119866)

= 119879sum

119866isinC

120572119866= 1 (11)



sum119866isinC 120572

119866= 1|aut (C)|





119901=2cup119896

119902=0119881119901119902

biconn let119881119899119896

119883=

119866 isin 119881119899119896


10

119883=



=

sum1198661015840isin119883119901119902 120590

1198661015840119866

1015840 with 1205901198661015840 isin Q and 119883

119901119902

sube 119881119901119902

119887119894119888119900119899119899 is such that



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin

Q119881119899119896


12057410

119888119900119899119899= 119866 119908ℎ119890119903119890 119866 = (1 0)

1205741119896

119888119900119899119899= 0 if 119896 gt 0

120574119899119896

119888119900119899119899=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1) 119903

120574119901119902

119887119894119888119900119899119899

119894(120574

119899minus119901+1119896minus119902

conn ))

(12)

Then 120574119899119896119888119900119899119899

= sum119866isin119881119899119896

119883

120572119866119866 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1| autC|







infin

119896=0S119896(119881) where





997888rarr S(119881)otimes119899


119899 1199041015840


1015840

119899)

997891997888rarr 11990411199041015840


1198991199041015840

119899

(13)




1

(a)

1 2

(b)

3

21

(c)


12isin S(119881)


12sdot 119877

13sdot 119877

23isin S(119881)

otimes3


otimes119899

119877119894119895= 1

otimes119894minus1



(14)


119894119895isin S(119881)

otimes119899 and 119877119894119895

isin S(119881)otimes1198991015840

with 119899 = 1198991015840



(ii) a tensor 119877119894119895


a graph on 1 119899


and 119864(119866) = 119894119896 119895119896119896=1119898



S(119881)otimes119899



119894119896119895119896



1119899isin S(119881)


119894119896119895119896

119878119866

1119899=

119898

prod

119896=1

119877119894119896119895119896

(15)


1015840



1015840

119878119866

120590(1)120590(119899)=

119898

prod

119896=1

119877120590(119894119896)120590(119895119896) (16)



1

3 4

2

(a)

5

3

4

1

2

(b)






5 3 997891rarr 4 4 997891rarr 2


1015840



119896=0S(119881)





119899) ∙ (119904

1015840


1015840

1198991015840)


119899otimes 119904

1015840


1015840

1198991015840

(18)

where 119904119894 1199041015840




1otimes 119904

2997891rarr 119904

2otimes 119904

1


otimes119896minus1



1015840 we define ∙119894119895


rarr S(119881)otimes119899+1198991015840



119899) ∙119894119895(1199041015840


1015840

1198991015840)

= (120591119899minus1

∘ sdot sdot sdot ∘ 120591119894) (119904


119899)


119895minus1) (119904

1015840


1015840

1198991015840)



119899otimes 119904

119894

otimes 1199041015840

119895otimes 119904

1015840


1199041015840


1015840

1198991015840

(19)

where 119904119894(resp 1199041015840

119895) means that 119904

119894(resp 1199041015840



isin S(119881)otimes119899 and 119878119866

1015840

11198991015840 isin

S(119881)otimes1198991015840


119878119866

1119899∙1198941198951198781198661015840

11198991015840 = 119878

119866

120590(1)120590(119899)sdot 1198781198661015840

1205901015840(1)120590

1015840(1198991015840) (20)

where 119878119866120590(1)120590(119899)

1198781198661015840

1205901015840(1)120590

1015840(1198991015840)isin S(119881)

otimes119899+1198991015840

120590(119896) = 119896 if 1 le 119896 lt

119894 120590(119894) = 119899 120590(119896) = 119896 minus 1 if 119894 lt 119896 le 119899 and 1205901015840

(119896) = 119896 + 119899 + 1

if 1 le 119896 lt 119895 1205901015840(119895) = 119899 + 1 1205901015840(119896) = 119896 + 119899 if 119895 lt 119896 le 1198991015840


1

3 41

32 2119866prime119866

(a)

5

6

1

4 3

2

119867

(b)



and 1198781198661015840

123isin S(119881)


=

119878119866

1234321198781198661015840

123isin S(119881)

otimes6


∙1198941198951198781198661015840



otimes119894minus1


S(119881)otimes119899




119894119895= sdot

119899∘ ∙

119894119895 S(119881)

otimes119899

times S(119881)otimes1198991015840

997888rarr S(119881)otimes119899+1198991015840

minus1

(21)

The tensor 1198781198661119899

1198941198951198781198661015840


1015840

minus 1


6 Linear Mappings


119899119896sub S(119881)


119899119896

biconn Let B =

B10⨁B

20⨁

infin

119899=3⨁

119896max(119899)

119896=1B

119899119896sub T(S(119881)) where 119896max(119899) =



Δ (1) = 1 otimes 1

Δ (119861119866

1119899) =

1

119899

119899

sum

119894=1

Δ119894(119861

119866

1119899) if 119899 gt 1

(22)


119866

1119899isin S(119881)



(i) 120590119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894

119895 + 1 if 119894 + 1 le 119895 le 119899

(ii) 120584119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894 minus 1

119895 + 1 if 119894 le 119895 le 119899

(23)


otimes119899


equation

Δ119894(119861

119866

1119899) = 119861

119866

120590119894(1)120590

119894(119899)

+ 119861119866

120584119894(1)120584

119894(119899)

= 119861119866

1119894+1119894+2119899+1+ 119861

119866

1119894119894+1119899+1

(24)


1119894119899+1(resp

119861119866

1119894+1119894+2119899+1





isin S(119881)otimes1198991015840


1015840


Δ119894(119861

119866

120587(1)120587(119899)) = 119861

119866

120584119894(120587(1))120584

119894(120587(119899))

(25)



Δ119894∘ Δ

119894= Δ

119894+1∘ Δ

119894 (26)


S(119881)otimes119899

rarr


and Δ119894 S(119881)

otimes119899+1



S(119881)otimes119899+1



119894 Δ119898

119894

S(119881)otimes119899


Δ1

119894= Δ

119894 (27)

Δ119898

119894= Δ

119894∘ Δ

119898minus1

119894 (28)

where Δ119894

S(119881)otimes119899+119898minus1






S(119881)otimes119899+119898minus1





= ⨁infin

120583=0B120583 where B0



otimes119899 and 119861119901sub S(119881)

otimes119901 by119860119899119894119895119861119901



119899 119887 isin 119861

119901) Also let B

2=

B20

and B119899= ⨁

119896max(119899)

119896=1B



119899

120583 as follows

Blowast

119899

120583

= ⋃


120583=119899+120583minus1

1198991

⋃

1198941=1

sdot sdot sdot

119899120583minus1

⋃

119894120583minus1=1

119899120583+sdotsdotsdot+119899

2minus120583+2

⋃

1198951=1

sdot sdot sdot

119899120583

⋃

119895120583minus1=1

B1198991

11989411198951

(B1198992

11989421198952


119894120583minus2119895120583minus2

(B119899120583minus1

119894120583minus1119895120583minus1

B119899120583

)) sdot sdot sdot)

(29)

Also define

B1015840

119899

lowast120583

= ⋃

120587isin119878119899

119878119866

120587(1)120587(119899)| 119878

119866

1119899isin B

lowast

119899

120583

(30)


and 119878119866


define

B120583

=

infin

⨁

119899=120583+1

B1015840

119899

lowast120583

(31)


otimes119899




prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886) (32)


119886(119899119886)isin S(119881)

otimes119899 119866119886is a


119861119866119886

1119899119886

isin S(119881)otimes119899119886 and 120590

119886 1 119899

119886 rarr 119883

119886sube 1 119899 is a


119899



Δ119894(

120583

prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886)) =

120583

prod

119886=1

Δ119894(119861

119866119886

120590119886(1)120590

119886(119899119886)) (33)










119866

1119899isin


119894119894+1119894+119899minus1= 119861

119866

120587119894(1)120587

119894(119899)




119894119894+1119894+119899minus1

with Δ119899minus1

119894

119861119866

119894119894+1119894+119899minus1∘ Δ

119899minus1

119894 S(119881)

otimes119901


(34)



∘ Δ119899minus1

119894produce







∘ Δ



120587119894(1)120587

119894(119899)

∘ Δ119894


123sdot1198611198623

345isin S(119881)


123= 119877

12sdot 119877

23sdot 119877

13isin

S(119881)otimes3 Let 119879


1198792

12=

11987712


34∘ Δ

3to119867 yields

11987734

∘ Δ3(119861

1198623

123sdot 119861

1198623

345)

= 11987734

sdot Δ3(119861

1198623

123) sdot Δ

3(119861

1198623

345)

= 11987734

sdot (1198611198623

123+ 119861

1198623

124) sdot (119861

1198623

356+ 119861

1198623

456)

= 11987734

sdot 1198611198623

123sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

123sdot 119861

1198623

456

+ 11987734

sdot 1198611198623

124sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

124sdot 119861

1198623

456

(35)



+ 2= 211987734 ∘ ( )Δ3



sharing a vertex








119899

119901=2cup119896

119902=0119881119901119902

biconnlet 119881119899119896

119883= 119866 isin 119881

119899119896


vention 11988110




1119901=

sum1198661015840isin119883119901119902 120590

11986610158401198611198661015840

1119901isin B

119901119902sub S(119881)


119901119902

sube

119881119901119902



1198661015840 gt 0 (ii)

sum1198661015840isinA 120590


define Ψ119899119896


Ψ10

= 1

Ψ1119896

= 0 if 119896 gt 0

(36)

Ψ119899119896

=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1)Φ119901119902

119894119894+1119894+119901minus1

sdotΔ119901minus1

119894(Ψ

119899minus119901+1119896minus119902

))

(37)

Then Ψ119899119896

= sum119866isin119881119899119896

119883

120572119866119878119866

1119899 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1|autC|



119881120583

119884= 119866 isin 119881

(119901minus1)120583+1119902120583


119866) = 120583

(38)





119901119902


119901119902

1119901=

sum1198661015840isin119884

12059011986610158401198611198661015840

1119901isin B

119901119902sub S(119881)



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin


1205950

= 1

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

(39)

Then 120595120583 = sum119866isin119881120583

119884

120572119866119878119866

1(119901minus1)120583+1with 120572





119866isinC 120572119866= 1|aut (C)|



(119901minus1)120583+1119902120583




120595120583

=

1

120583

times sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(119894


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(40)



1205951

= 120601119901119902


119901119902

1119901 (41)


120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


119894(120595

1198941


)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941


119894(120595

1198942


)

+ sdot sdot sdot +


sum


Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119903cutvertex

= 4 + 8 + 4)(



=)(119888




(a) 119881(119866119910) = 119881(119866)

(b) 119864(119866119910) = 119864(119866)

(c) 119897119866119910

(119907119894) = 119897

119866(119907119894) cup 119871

1015840(119894) for all 119894 = 1 119899 and119907119894isin 119881(119866)


119899119896119871 by line-arity







=

sum1198661015840isin119881119901119902

119887119894119888119900119899119899

1205901198661015840119866


classA isin E(119881119901119902



1198661015840isinA 120590



isin

Q119881119899119896119871


12057310119871

119888119900119899119899= 119866 119908ℎ119890119903119890 119866 = (1 0) 119897

119866(1) = 119871

1205731119896119871

119888119900119899119899= 0 119894119891 119896 gt 0

120573119899119896119871

119888119900119899119899=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1) 119903

120573119901119902

119887119894119888119900119899119899

119894(120573

119899minus119901+1119896minus119902119871

119888119900119899119899))

(8)

Then 120573119899119896119871119888119900119899119899

= sum119866isin119881119899119896119871

119888119900119899119899

120572119866119866 with 120572




119866gt 0 (ii) sum

119866isinC 120572119866= 1|autC|



119866isin119881119899119896119871



119866gt 0




biconn1205901198661015840119866

1015840 with


119901119902

biconn119894

read as

119903

120573119901119902

biconn119894

= sum

1198661015840isin119881119901119902

biconn

12059011986610158401199031198661015840

119894 (9)


120573119901119902

biconn119894


isin 119881119899minus119901+1119896minus119902119871



119866lowast is the



119907isin119881(119866)119897119866(119907)



notin


119899minus119901+1119896minus119902119871






cong

119888119866(119866) and 120584

119867lowast gt 0 Let 119907

119895isin 119881(119867

lowast




isin 119881119901119902




119867gt 0


119888119900119899119899= sum

119866isin119881119899119896119871

119888119900119899119899

120572119866119866 be




sum119866isinC 120572

119866= 1




119866gt 0 where 120572


in 120573119899119896119871







1199011199021198711015840


= cup119907isin119881(

119866)119897119866(119907) Let 1198981015840

= card119864(119866) so that 1198981015840

=


1015840


119899minus119901+1119896minus119902119871


119888119866(119866)



conn120584119866lowast119866


that 119867 cong 119888119866(119866) and 120584



119866lowast = 1 Now let 119907



119901119902


119901119902


1015840




1015840 with

119897119888119866(119866)


1198661015840

119895(119867)



1198661015840



= 120584119867


1198661015840



119899minus119901+1119896minus119902119871


sum

119866isinC

120572lowast


119866lowastisinD

120584119866lowast

= |autA|

(10)


119866isinC 120572119866is 1198981015840



119866isinC 120572119866





= 0 Then 120573119899119896119871cup1198711015840

119888119900119899119899= 120585

1198711015840(120573

119899119896119871

119888119900119899119899)

Proof Let 1205851198711015840 119881

119899119896119871

rarr 119881119899119896119871cup119871

1015840

and 1205851198711015840 119881

119899minus119901+1119896minus119902119871

rarr

119881119899minus119901+1119896minus119902119871cup119871

1015840


1198711015840 ∘ 119903

120573119901119902

biconn119894

= 119903

120573119901119902

biconn119894

∘1205851198711015840


119888119900119899119899= sum

119866isin119881119899119896119871

119888119900119899119899


(8) Let C isin E(119881119899119896119871119888119900119899119899


119866= 1|aut (C)|



119866(119881

1015840


1015840



such that 119897|119881(119866)119881

1015840 = 119897119866|119881(119866)119881

1015840 and 119897(119907) = 0 for all 119907 isin 1198811015840


isin 119881119899119896119871cup119871

1015840


1015840


1015840

conn = sum1198661015840isin119881119899119896119871cup119871

1015840

conn1205841198661015840119866


3 sum1198661015840isinD 120584



119871cup1198711015840

119895 = 1 119879 such that119897119895|119881(119866)119881

1015840 = 119897119866|119881(119866)119881

1015840 119897119895(119907) = 0 for all 119907 isin 119881

1015840 and 119866119895isin D


119895 Clearly

120584119866119895

= 120572119866gt 0 where 120584

119866119895


119899119896119871cup1198711015840

conn Define 119891

119895 119866 997891rarr 119866



12057310conn =

12057320conn =

12057330conn =

12057340conn =

12057341conn =

12057331conn =

12057340conn =

1

2

1

3

1

8

+1

2

+1

2

+1

4

+1

3

1

2

1

2

1

2




sum

1198661015840isinD

1205841198661015840 =

119879

sum

119895=1

sum

119866isinC

120584119891119895(119866)

= 119879sum

119866isinC

120572119866= 1 (11)



sum119866isinC 120572

119866= 1|aut (C)|





119901=2cup119896

119902=0119881119901119902

biconn let119881119899119896

119883=

119866 isin 119881119899119896


10

119883=



=

sum1198661015840isin119883119901119902 120590

1198661015840119866

1015840 with 1205901198661015840 isin Q and 119883

119901119902

sube 119881119901119902

119887119894119888119900119899119899 is such that



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin

Q119881119899119896


12057410

119888119900119899119899= 119866 119908ℎ119890119903119890 119866 = (1 0)

1205741119896

119888119900119899119899= 0 if 119896 gt 0

120574119899119896

119888119900119899119899=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1) 119903

120574119901119902

119887119894119888119900119899119899

119894(120574

119899minus119901+1119896minus119902

conn ))

(12)

Then 120574119899119896119888119900119899119899

= sum119866isin119881119899119896

119883

120572119866119866 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1| autC|







infin

119896=0S119896(119881) where





997888rarr S(119881)otimes119899


119899 1199041015840


1015840

119899)

997891997888rarr 11990411199041015840


1198991199041015840

119899

(13)




1

(a)

1 2

(b)

3

21

(c)


12isin S(119881)


12sdot 119877

13sdot 119877

23isin S(119881)

otimes3


otimes119899

119877119894119895= 1

otimes119894minus1



(14)


119894119895isin S(119881)

otimes119899 and 119877119894119895

isin S(119881)otimes1198991015840

with 119899 = 1198991015840



(ii) a tensor 119877119894119895


a graph on 1 119899


and 119864(119866) = 119894119896 119895119896119896=1119898



S(119881)otimes119899



119894119896119895119896



1119899isin S(119881)


119894119896119895119896

119878119866

1119899=

119898

prod

119896=1

119877119894119896119895119896

(15)


1015840



1015840

119878119866

120590(1)120590(119899)=

119898

prod

119896=1

119877120590(119894119896)120590(119895119896) (16)



1

3 4

2

(a)

5

3

4

1

2

(b)






5 3 997891rarr 4 4 997891rarr 2


1015840



119896=0S(119881)





119899) ∙ (119904

1015840


1015840

1198991015840)


119899otimes 119904

1015840


1015840

1198991015840

(18)

where 119904119894 1199041015840




1otimes 119904

2997891rarr 119904

2otimes 119904

1


otimes119896minus1



1015840 we define ∙119894119895


rarr S(119881)otimes119899+1198991015840



119899) ∙119894119895(1199041015840


1015840

1198991015840)

= (120591119899minus1

∘ sdot sdot sdot ∘ 120591119894) (119904


119899)


119895minus1) (119904

1015840


1015840

1198991015840)



119899otimes 119904

119894

otimes 1199041015840

119895otimes 119904

1015840


1199041015840


1015840

1198991015840

(19)

where 119904119894(resp 1199041015840

119895) means that 119904

119894(resp 1199041015840



isin S(119881)otimes119899 and 119878119866

1015840

11198991015840 isin

S(119881)otimes1198991015840


119878119866

1119899∙1198941198951198781198661015840

11198991015840 = 119878

119866

120590(1)120590(119899)sdot 1198781198661015840

1205901015840(1)120590

1015840(1198991015840) (20)

where 119878119866120590(1)120590(119899)

1198781198661015840

1205901015840(1)120590

1015840(1198991015840)isin S(119881)

otimes119899+1198991015840

120590(119896) = 119896 if 1 le 119896 lt

119894 120590(119894) = 119899 120590(119896) = 119896 minus 1 if 119894 lt 119896 le 119899 and 1205901015840

(119896) = 119896 + 119899 + 1

if 1 le 119896 lt 119895 1205901015840(119895) = 119899 + 1 1205901015840(119896) = 119896 + 119899 if 119895 lt 119896 le 1198991015840


1

3 41

32 2119866prime119866

(a)

5

6

1

4 3

2

119867

(b)



and 1198781198661015840

123isin S(119881)


=

119878119866

1234321198781198661015840

123isin S(119881)

otimes6


∙1198941198951198781198661015840



otimes119894minus1


S(119881)otimes119899




119894119895= sdot

119899∘ ∙

119894119895 S(119881)

otimes119899

times S(119881)otimes1198991015840

997888rarr S(119881)otimes119899+1198991015840

minus1

(21)

The tensor 1198781198661119899

1198941198951198781198661015840


1015840

minus 1


6 Linear Mappings


119899119896sub S(119881)


119899119896

biconn Let B =

B10⨁B

20⨁

infin

119899=3⨁

119896max(119899)

119896=1B

119899119896sub T(S(119881)) where 119896max(119899) =



Δ (1) = 1 otimes 1

Δ (119861119866

1119899) =

1

119899

119899

sum

119894=1

Δ119894(119861

119866

1119899) if 119899 gt 1

(22)


119866

1119899isin S(119881)



(i) 120590119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894

119895 + 1 if 119894 + 1 le 119895 le 119899

(ii) 120584119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894 minus 1

119895 + 1 if 119894 le 119895 le 119899

(23)


otimes119899


equation

Δ119894(119861

119866

1119899) = 119861

119866

120590119894(1)120590

119894(119899)

+ 119861119866

120584119894(1)120584

119894(119899)

= 119861119866

1119894+1119894+2119899+1+ 119861

119866

1119894119894+1119899+1

(24)


1119894119899+1(resp

119861119866

1119894+1119894+2119899+1





isin S(119881)otimes1198991015840


1015840


Δ119894(119861

119866

120587(1)120587(119899)) = 119861

119866

120584119894(120587(1))120584

119894(120587(119899))

(25)



Δ119894∘ Δ

119894= Δ

119894+1∘ Δ

119894 (26)


S(119881)otimes119899

rarr


and Δ119894 S(119881)

otimes119899+1



S(119881)otimes119899+1



119894 Δ119898

119894

S(119881)otimes119899


Δ1

119894= Δ

119894 (27)

Δ119898

119894= Δ

119894∘ Δ

119898minus1

119894 (28)

where Δ119894

S(119881)otimes119899+119898minus1






S(119881)otimes119899+119898minus1





= ⨁infin

120583=0B120583 where B0



otimes119899 and 119861119901sub S(119881)

otimes119901 by119860119899119894119895119861119901



119899 119887 isin 119861

119901) Also let B

2=

B20

and B119899= ⨁

119896max(119899)

119896=1B



119899

120583 as follows

Blowast

119899

120583

= ⋃


120583=119899+120583minus1

1198991

⋃

1198941=1

sdot sdot sdot

119899120583minus1

⋃

119894120583minus1=1

119899120583+sdotsdotsdot+119899

2minus120583+2

⋃

1198951=1

sdot sdot sdot

119899120583

⋃

119895120583minus1=1

B1198991

11989411198951

(B1198992

11989421198952


119894120583minus2119895120583minus2

(B119899120583minus1

119894120583minus1119895120583minus1

B119899120583

)) sdot sdot sdot)

(29)

Also define

B1015840

119899

lowast120583

= ⋃

120587isin119878119899

119878119866

120587(1)120587(119899)| 119878

119866

1119899isin B

lowast

119899

120583

(30)


and 119878119866


define

B120583

=

infin

⨁

119899=120583+1

B1015840

119899

lowast120583

(31)


otimes119899




prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886) (32)


119886(119899119886)isin S(119881)

otimes119899 119866119886is a


119861119866119886

1119899119886

isin S(119881)otimes119899119886 and 120590

119886 1 119899

119886 rarr 119883

119886sube 1 119899 is a


119899



Δ119894(

120583

prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886)) =

120583

prod

119886=1

Δ119894(119861

119866119886

120590119886(1)120590

119886(119899119886)) (33)










119866

1119899isin


119894119894+1119894+119899minus1= 119861

119866

120587119894(1)120587

119894(119899)




119894119894+1119894+119899minus1

with Δ119899minus1

119894

119861119866

119894119894+1119894+119899minus1∘ Δ

119899minus1

119894 S(119881)

otimes119901


(34)



∘ Δ119899minus1

119894produce







∘ Δ



120587119894(1)120587

119894(119899)

∘ Δ119894


123sdot1198611198623

345isin S(119881)


123= 119877

12sdot 119877

23sdot 119877

13isin

S(119881)otimes3 Let 119879


1198792

12=

11987712


34∘ Δ

3to119867 yields

11987734

∘ Δ3(119861

1198623

123sdot 119861

1198623

345)

= 11987734

sdot Δ3(119861

1198623

123) sdot Δ

3(119861

1198623

345)

= 11987734

sdot (1198611198623

123+ 119861

1198623

124) sdot (119861

1198623

356+ 119861

1198623

456)

= 11987734

sdot 1198611198623

123sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

123sdot 119861

1198623

456

+ 11987734

sdot 1198611198623

124sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

124sdot 119861

1198623

456

(35)



+ 2= 211987734 ∘ ( )Δ3



sharing a vertex








119899

119901=2cup119896

119902=0119881119901119902

biconnlet 119881119899119896

119883= 119866 isin 119881

119899119896


vention 11988110




1119901=

sum1198661015840isin119883119901119902 120590

11986610158401198611198661015840

1119901isin B

119901119902sub S(119881)


119901119902

sube

119881119901119902



1198661015840 gt 0 (ii)

sum1198661015840isinA 120590


define Ψ119899119896


Ψ10

= 1

Ψ1119896

= 0 if 119896 gt 0

(36)

Ψ119899119896

=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1)Φ119901119902

119894119894+1119894+119901minus1

sdotΔ119901minus1

119894(Ψ

119899minus119901+1119896minus119902

))

(37)

Then Ψ119899119896

= sum119866isin119881119899119896

119883

120572119866119878119866

1119899 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1|autC|



119881120583

119884= 119866 isin 119881

(119901minus1)120583+1119902120583


119866) = 120583

(38)





119901119902


119901119902

1119901=

sum1198661015840isin119884

12059011986610158401198611198661015840

1119901isin B

119901119902sub S(119881)



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin


1205950

= 1

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

(39)

Then 120595120583 = sum119866isin119881120583

119884

120572119866119878119866

1(119901minus1)120583+1with 120572





119866isinC 120572119866= 1|aut (C)|



(119901minus1)120583+1119902120583




120595120583

=

1

120583

times sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(119894


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(40)



1205951

= 120601119901119902


119901119902

1119901 (41)


120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


119894(120595

1198941


)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941


119894(120595

1198942


)

+ sdot sdot sdot +


sum


Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











notin


119899minus119901+1119896minus119902119871






cong

119888119866(119866) and 120584

119867lowast gt 0 Let 119907

119895isin 119881(119867

lowast




isin 119881119901119902




119867gt 0


119888119900119899119899= sum

119866isin119881119899119896119871

119888119900119899119899

120572119866119866 be




sum119866isinC 120572

119866= 1




119866gt 0 where 120572


in 120573119899119896119871







1199011199021198711015840


= cup119907isin119881(

119866)119897119866(119907) Let 1198981015840

= card119864(119866) so that 1198981015840

=


1015840


119899minus119901+1119896minus119902119871


119888119866(119866)



conn120584119866lowast119866


that 119867 cong 119888119866(119866) and 120584



119866lowast = 1 Now let 119907



119901119902


119901119902


1015840




1015840 with

119897119888119866(119866)


1198661015840

119895(119867)



1198661015840



= 120584119867


1198661015840



119899minus119901+1119896minus119902119871


sum

119866isinC

120572lowast


119866lowastisinD

120584119866lowast

= |autA|

(10)


119866isinC 120572119866is 1198981015840



119866isinC 120572119866





= 0 Then 120573119899119896119871cup1198711015840

119888119900119899119899= 120585

1198711015840(120573

119899119896119871

119888119900119899119899)

Proof Let 1205851198711015840 119881

119899119896119871

rarr 119881119899119896119871cup119871

1015840

and 1205851198711015840 119881

119899minus119901+1119896minus119902119871

rarr

119881119899minus119901+1119896minus119902119871cup119871

1015840


1198711015840 ∘ 119903

120573119901119902

biconn119894

= 119903

120573119901119902

biconn119894

∘1205851198711015840


119888119900119899119899= sum

119866isin119881119899119896119871

119888119900119899119899


(8) Let C isin E(119881119899119896119871119888119900119899119899


119866= 1|aut (C)|



119866(119881

1015840


1015840



such that 119897|119881(119866)119881

1015840 = 119897119866|119881(119866)119881

1015840 and 119897(119907) = 0 for all 119907 isin 1198811015840


isin 119881119899119896119871cup119871

1015840


1015840


1015840

conn = sum1198661015840isin119881119899119896119871cup119871

1015840

conn1205841198661015840119866


3 sum1198661015840isinD 120584



119871cup1198711015840

119895 = 1 119879 such that119897119895|119881(119866)119881

1015840 = 119897119866|119881(119866)119881

1015840 119897119895(119907) = 0 for all 119907 isin 119881

1015840 and 119866119895isin D


119895 Clearly

120584119866119895

= 120572119866gt 0 where 120584

119866119895


119899119896119871cup1198711015840

conn Define 119891

119895 119866 997891rarr 119866



12057310conn =

12057320conn =

12057330conn =

12057340conn =

12057341conn =

12057331conn =

12057340conn =

1

2

1

3

1

8

+1

2

+1

2

+1

4

+1

3

1

2

1

2

1

2




sum

1198661015840isinD

1205841198661015840 =

119879

sum

119895=1

sum

119866isinC

120584119891119895(119866)

= 119879sum

119866isinC

120572119866= 1 (11)



sum119866isinC 120572

119866= 1|aut (C)|





119901=2cup119896

119902=0119881119901119902

biconn let119881119899119896

119883=

119866 isin 119881119899119896


10

119883=



=

sum1198661015840isin119883119901119902 120590

1198661015840119866

1015840 with 1205901198661015840 isin Q and 119883

119901119902

sube 119881119901119902

119887119894119888119900119899119899 is such that



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin

Q119881119899119896


12057410

119888119900119899119899= 119866 119908ℎ119890119903119890 119866 = (1 0)

1205741119896

119888119900119899119899= 0 if 119896 gt 0

120574119899119896

119888119900119899119899=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1) 119903

120574119901119902

119887119894119888119900119899119899

119894(120574

119899minus119901+1119896minus119902

conn ))

(12)

Then 120574119899119896119888119900119899119899

= sum119866isin119881119899119896

119883

120572119866119866 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1| autC|







infin

119896=0S119896(119881) where





997888rarr S(119881)otimes119899


119899 1199041015840


1015840

119899)

997891997888rarr 11990411199041015840


1198991199041015840

119899

(13)




1

(a)

1 2

(b)

3

21

(c)


12isin S(119881)


12sdot 119877

13sdot 119877

23isin S(119881)

otimes3


otimes119899

119877119894119895= 1

otimes119894minus1



(14)


119894119895isin S(119881)

otimes119899 and 119877119894119895

isin S(119881)otimes1198991015840

with 119899 = 1198991015840



(ii) a tensor 119877119894119895


a graph on 1 119899


and 119864(119866) = 119894119896 119895119896119896=1119898



S(119881)otimes119899



119894119896119895119896



1119899isin S(119881)


119894119896119895119896

119878119866

1119899=

119898

prod

119896=1

119877119894119896119895119896

(15)


1015840



1015840

119878119866

120590(1)120590(119899)=

119898

prod

119896=1

119877120590(119894119896)120590(119895119896) (16)



1

3 4

2

(a)

5

3

4

1

2

(b)






5 3 997891rarr 4 4 997891rarr 2


1015840



119896=0S(119881)





119899) ∙ (119904

1015840


1015840

1198991015840)


119899otimes 119904

1015840


1015840

1198991015840

(18)

where 119904119894 1199041015840




1otimes 119904

2997891rarr 119904

2otimes 119904

1


otimes119896minus1



1015840 we define ∙119894119895


rarr S(119881)otimes119899+1198991015840



119899) ∙119894119895(1199041015840


1015840

1198991015840)

= (120591119899minus1

∘ sdot sdot sdot ∘ 120591119894) (119904


119899)


119895minus1) (119904

1015840


1015840

1198991015840)



119899otimes 119904

119894

otimes 1199041015840

119895otimes 119904

1015840


1199041015840


1015840

1198991015840

(19)

where 119904119894(resp 1199041015840

119895) means that 119904

119894(resp 1199041015840



isin S(119881)otimes119899 and 119878119866

1015840

11198991015840 isin

S(119881)otimes1198991015840


119878119866

1119899∙1198941198951198781198661015840

11198991015840 = 119878

119866

120590(1)120590(119899)sdot 1198781198661015840

1205901015840(1)120590

1015840(1198991015840) (20)

where 119878119866120590(1)120590(119899)

1198781198661015840

1205901015840(1)120590

1015840(1198991015840)isin S(119881)

otimes119899+1198991015840

120590(119896) = 119896 if 1 le 119896 lt

119894 120590(119894) = 119899 120590(119896) = 119896 minus 1 if 119894 lt 119896 le 119899 and 1205901015840

(119896) = 119896 + 119899 + 1

if 1 le 119896 lt 119895 1205901015840(119895) = 119899 + 1 1205901015840(119896) = 119896 + 119899 if 119895 lt 119896 le 1198991015840


1

3 41

32 2119866prime119866

(a)

5

6

1

4 3

2

119867

(b)



and 1198781198661015840

123isin S(119881)


=

119878119866

1234321198781198661015840

123isin S(119881)

otimes6


∙1198941198951198781198661015840



otimes119894minus1


S(119881)otimes119899




119894119895= sdot

119899∘ ∙

119894119895 S(119881)

otimes119899

times S(119881)otimes1198991015840

997888rarr S(119881)otimes119899+1198991015840

minus1

(21)

The tensor 1198781198661119899

1198941198951198781198661015840


1015840

minus 1


6 Linear Mappings


119899119896sub S(119881)


119899119896

biconn Let B =

B10⨁B

20⨁

infin

119899=3⨁

119896max(119899)

119896=1B

119899119896sub T(S(119881)) where 119896max(119899) =



Δ (1) = 1 otimes 1

Δ (119861119866

1119899) =

1

119899

119899

sum

119894=1

Δ119894(119861

119866

1119899) if 119899 gt 1

(22)


119866

1119899isin S(119881)



(i) 120590119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894

119895 + 1 if 119894 + 1 le 119895 le 119899

(ii) 120584119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894 minus 1

119895 + 1 if 119894 le 119895 le 119899

(23)


otimes119899


equation

Δ119894(119861

119866

1119899) = 119861

119866

120590119894(1)120590

119894(119899)

+ 119861119866

120584119894(1)120584

119894(119899)

= 119861119866

1119894+1119894+2119899+1+ 119861

119866

1119894119894+1119899+1

(24)


1119894119899+1(resp

119861119866

1119894+1119894+2119899+1





isin S(119881)otimes1198991015840


1015840


Δ119894(119861

119866

120587(1)120587(119899)) = 119861

119866

120584119894(120587(1))120584

119894(120587(119899))

(25)



Δ119894∘ Δ

119894= Δ

119894+1∘ Δ

119894 (26)


S(119881)otimes119899

rarr


and Δ119894 S(119881)

otimes119899+1



S(119881)otimes119899+1



119894 Δ119898

119894

S(119881)otimes119899


Δ1

119894= Δ

119894 (27)

Δ119898

119894= Δ

119894∘ Δ

119898minus1

119894 (28)

where Δ119894

S(119881)otimes119899+119898minus1






S(119881)otimes119899+119898minus1





= ⨁infin

120583=0B120583 where B0



otimes119899 and 119861119901sub S(119881)

otimes119901 by119860119899119894119895119861119901



119899 119887 isin 119861

119901) Also let B

2=

B20

and B119899= ⨁

119896max(119899)

119896=1B



119899

120583 as follows

Blowast

119899

120583

= ⋃


120583=119899+120583minus1

1198991

⋃

1198941=1

sdot sdot sdot

119899120583minus1

⋃

119894120583minus1=1

119899120583+sdotsdotsdot+119899

2minus120583+2

⋃

1198951=1

sdot sdot sdot

119899120583

⋃

119895120583minus1=1

B1198991

11989411198951

(B1198992

11989421198952


119894120583minus2119895120583minus2

(B119899120583minus1

119894120583minus1119895120583minus1

B119899120583

)) sdot sdot sdot)

(29)

Also define

B1015840

119899

lowast120583

= ⋃

120587isin119878119899

119878119866

120587(1)120587(119899)| 119878

119866

1119899isin B

lowast

119899

120583

(30)


and 119878119866


define

B120583

=

infin

⨁

119899=120583+1

B1015840

119899

lowast120583

(31)


otimes119899




prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886) (32)


119886(119899119886)isin S(119881)

otimes119899 119866119886is a


119861119866119886

1119899119886

isin S(119881)otimes119899119886 and 120590

119886 1 119899

119886 rarr 119883

119886sube 1 119899 is a


119899



Δ119894(

120583

prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886)) =

120583

prod

119886=1

Δ119894(119861

119866119886

120590119886(1)120590

119886(119899119886)) (33)










119866

1119899isin


119894119894+1119894+119899minus1= 119861

119866

120587119894(1)120587

119894(119899)




119894119894+1119894+119899minus1

with Δ119899minus1

119894

119861119866

119894119894+1119894+119899minus1∘ Δ

119899minus1

119894 S(119881)

otimes119901


(34)



∘ Δ119899minus1

119894produce







∘ Δ



120587119894(1)120587

119894(119899)

∘ Δ119894


123sdot1198611198623

345isin S(119881)


123= 119877

12sdot 119877

23sdot 119877

13isin

S(119881)otimes3 Let 119879


1198792

12=

11987712


34∘ Δ

3to119867 yields

11987734

∘ Δ3(119861

1198623

123sdot 119861

1198623

345)

= 11987734

sdot Δ3(119861

1198623

123) sdot Δ

3(119861

1198623

345)

= 11987734

sdot (1198611198623

123+ 119861

1198623

124) sdot (119861

1198623

356+ 119861

1198623

456)

= 11987734

sdot 1198611198623

123sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

123sdot 119861

1198623

456

+ 11987734

sdot 1198611198623

124sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

124sdot 119861

1198623

456

(35)



+ 2= 211987734 ∘ ( )Δ3



sharing a vertex








119899

119901=2cup119896

119902=0119881119901119902

biconnlet 119881119899119896

119883= 119866 isin 119881

119899119896


vention 11988110




1119901=

sum1198661015840isin119883119901119902 120590

11986610158401198611198661015840

1119901isin B

119901119902sub S(119881)


119901119902

sube

119881119901119902



1198661015840 gt 0 (ii)

sum1198661015840isinA 120590


define Ψ119899119896


Ψ10

= 1

Ψ1119896

= 0 if 119896 gt 0

(36)

Ψ119899119896

=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1)Φ119901119902

119894119894+1119894+119901minus1

sdotΔ119901minus1

119894(Ψ

119899minus119901+1119896minus119902

))

(37)

Then Ψ119899119896

= sum119866isin119881119899119896

119883

120572119866119878119866

1119899 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1|autC|



119881120583

119884= 119866 isin 119881

(119901minus1)120583+1119902120583


119866) = 120583

(38)





119901119902


119901119902

1119901=

sum1198661015840isin119884

12059011986610158401198611198661015840

1119901isin B

119901119902sub S(119881)



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin


1205950

= 1

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

(39)

Then 120595120583 = sum119866isin119881120583

119884

120572119866119878119866

1(119901minus1)120583+1with 120572





119866isinC 120572119866= 1|aut (C)|



(119901minus1)120583+1119902120583




120595120583

=

1

120583

times sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(119894


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(40)



1205951

= 120601119901119902


119901119902

1119901 (41)


120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


119894(120595

1198941


)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941


119894(120595

1198942


)

+ sdot sdot sdot +


sum


Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12057310conn =

12057320conn =

12057330conn =

12057340conn =

12057341conn =

12057331conn =

12057340conn =

1

2

1

3

1

8

+1

2

+1

2

+1

4

+1

3

1

2

1

2

1

2




sum

1198661015840isinD

1205841198661015840 =

119879

sum

119895=1

sum

119866isinC

120584119891119895(119866)

= 119879sum

119866isinC

120572119866= 1 (11)



sum119866isinC 120572

119866= 1|aut (C)|





119901=2cup119896

119902=0119881119901119902

biconn let119881119899119896

119883=

119866 isin 119881119899119896


10

119883=



=

sum1198661015840isin119883119901119902 120590

1198661015840119866

1015840 with 1205901198661015840 isin Q and 119883

119901119902

sube 119881119901119902

119887119894119888119900119899119899 is such that



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin

Q119881119899119896


12057410

119888119900119899119899= 119866 119908ℎ119890119903119890 119866 = (1 0)

1205741119896

119888119900119899119899= 0 if 119896 gt 0

120574119899119896

119888119900119899119899=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1) 119903

120574119901119902

119887119894119888119900119899119899

119894(120574

119899minus119901+1119896minus119902

conn ))

(12)

Then 120574119899119896119888119900119899119899

= sum119866isin119881119899119896

119883

120572119866119866 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1| autC|







infin

119896=0S119896(119881) where





997888rarr S(119881)otimes119899


119899 1199041015840


1015840

119899)

997891997888rarr 11990411199041015840


1198991199041015840

119899

(13)




1

(a)

1 2

(b)

3

21

(c)


12isin S(119881)


12sdot 119877

13sdot 119877

23isin S(119881)

otimes3


otimes119899

119877119894119895= 1

otimes119894minus1



(14)


119894119895isin S(119881)

otimes119899 and 119877119894119895

isin S(119881)otimes1198991015840

with 119899 = 1198991015840



(ii) a tensor 119877119894119895


a graph on 1 119899


and 119864(119866) = 119894119896 119895119896119896=1119898



S(119881)otimes119899



119894119896119895119896



1119899isin S(119881)


119894119896119895119896

119878119866

1119899=

119898

prod

119896=1

119877119894119896119895119896

(15)


1015840



1015840

119878119866

120590(1)120590(119899)=

119898

prod

119896=1

119877120590(119894119896)120590(119895119896) (16)



1

3 4

2

(a)

5

3

4

1

2

(b)






5 3 997891rarr 4 4 997891rarr 2


1015840



119896=0S(119881)





119899) ∙ (119904

1015840


1015840

1198991015840)


119899otimes 119904

1015840


1015840

1198991015840

(18)

where 119904119894 1199041015840




1otimes 119904

2997891rarr 119904

2otimes 119904

1


otimes119896minus1



1015840 we define ∙119894119895


rarr S(119881)otimes119899+1198991015840



119899) ∙119894119895(1199041015840


1015840

1198991015840)

= (120591119899minus1

∘ sdot sdot sdot ∘ 120591119894) (119904


119899)


119895minus1) (119904

1015840


1015840

1198991015840)



119899otimes 119904

119894

otimes 1199041015840

119895otimes 119904

1015840


1199041015840


1015840

1198991015840

(19)

where 119904119894(resp 1199041015840

119895) means that 119904

119894(resp 1199041015840



isin S(119881)otimes119899 and 119878119866

1015840

11198991015840 isin

S(119881)otimes1198991015840


119878119866

1119899∙1198941198951198781198661015840

11198991015840 = 119878

119866

120590(1)120590(119899)sdot 1198781198661015840

1205901015840(1)120590

1015840(1198991015840) (20)

where 119878119866120590(1)120590(119899)

1198781198661015840

1205901015840(1)120590

1015840(1198991015840)isin S(119881)

otimes119899+1198991015840

120590(119896) = 119896 if 1 le 119896 lt

119894 120590(119894) = 119899 120590(119896) = 119896 minus 1 if 119894 lt 119896 le 119899 and 1205901015840

(119896) = 119896 + 119899 + 1

if 1 le 119896 lt 119895 1205901015840(119895) = 119899 + 1 1205901015840(119896) = 119896 + 119899 if 119895 lt 119896 le 1198991015840


1

3 41

32 2119866prime119866

(a)

5

6

1

4 3

2

119867

(b)



and 1198781198661015840

123isin S(119881)


=

119878119866

1234321198781198661015840

123isin S(119881)

otimes6


∙1198941198951198781198661015840



otimes119894minus1


S(119881)otimes119899




119894119895= sdot

119899∘ ∙

119894119895 S(119881)

otimes119899

times S(119881)otimes1198991015840

997888rarr S(119881)otimes119899+1198991015840

minus1

(21)

The tensor 1198781198661119899

1198941198951198781198661015840


1015840

minus 1


6 Linear Mappings


119899119896sub S(119881)


119899119896

biconn Let B =

B10⨁B

20⨁

infin

119899=3⨁

119896max(119899)

119896=1B

119899119896sub T(S(119881)) where 119896max(119899) =



Δ (1) = 1 otimes 1

Δ (119861119866

1119899) =

1

119899

119899

sum

119894=1

Δ119894(119861

119866

1119899) if 119899 gt 1

(22)


119866

1119899isin S(119881)



(i) 120590119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894

119895 + 1 if 119894 + 1 le 119895 le 119899

(ii) 120584119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894 minus 1

119895 + 1 if 119894 le 119895 le 119899

(23)


otimes119899


equation

Δ119894(119861

119866

1119899) = 119861

119866

120590119894(1)120590

119894(119899)

+ 119861119866

120584119894(1)120584

119894(119899)

= 119861119866

1119894+1119894+2119899+1+ 119861

119866

1119894119894+1119899+1

(24)


1119894119899+1(resp

119861119866

1119894+1119894+2119899+1





isin S(119881)otimes1198991015840


1015840


Δ119894(119861

119866

120587(1)120587(119899)) = 119861

119866

120584119894(120587(1))120584

119894(120587(119899))

(25)



Δ119894∘ Δ

119894= Δ

119894+1∘ Δ

119894 (26)


S(119881)otimes119899

rarr


and Δ119894 S(119881)

otimes119899+1



S(119881)otimes119899+1



119894 Δ119898

119894

S(119881)otimes119899


Δ1

119894= Δ

119894 (27)

Δ119898

119894= Δ

119894∘ Δ

119898minus1

119894 (28)

where Δ119894

S(119881)otimes119899+119898minus1






S(119881)otimes119899+119898minus1





= ⨁infin

120583=0B120583 where B0



otimes119899 and 119861119901sub S(119881)

otimes119901 by119860119899119894119895119861119901



119899 119887 isin 119861

119901) Also let B

2=

B20

and B119899= ⨁

119896max(119899)

119896=1B



119899

120583 as follows

Blowast

119899

120583

= ⋃


120583=119899+120583minus1

1198991

⋃

1198941=1

sdot sdot sdot

119899120583minus1

⋃

119894120583minus1=1

119899120583+sdotsdotsdot+119899

2minus120583+2

⋃

1198951=1

sdot sdot sdot

119899120583

⋃

119895120583minus1=1

B1198991

11989411198951

(B1198992

11989421198952


119894120583minus2119895120583minus2

(B119899120583minus1

119894120583minus1119895120583minus1

B119899120583

)) sdot sdot sdot)

(29)

Also define

B1015840

119899

lowast120583

= ⋃

120587isin119878119899

119878119866

120587(1)120587(119899)| 119878

119866

1119899isin B

lowast

119899

120583

(30)


and 119878119866


define

B120583

=

infin

⨁

119899=120583+1

B1015840

119899

lowast120583

(31)


otimes119899




prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886) (32)


119886(119899119886)isin S(119881)

otimes119899 119866119886is a


119861119866119886

1119899119886

isin S(119881)otimes119899119886 and 120590

119886 1 119899

119886 rarr 119883

119886sube 1 119899 is a


119899



Δ119894(

120583

prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886)) =

120583

prod

119886=1

Δ119894(119861

119866119886

120590119886(1)120590

119886(119899119886)) (33)










119866

1119899isin


119894119894+1119894+119899minus1= 119861

119866

120587119894(1)120587

119894(119899)




119894119894+1119894+119899minus1

with Δ119899minus1

119894

119861119866

119894119894+1119894+119899minus1∘ Δ

119899minus1

119894 S(119881)

otimes119901


(34)



∘ Δ119899minus1

119894produce







∘ Δ



120587119894(1)120587

119894(119899)

∘ Δ119894


123sdot1198611198623

345isin S(119881)


123= 119877

12sdot 119877

23sdot 119877

13isin

S(119881)otimes3 Let 119879


1198792

12=

11987712


34∘ Δ

3to119867 yields

11987734

∘ Δ3(119861

1198623

123sdot 119861

1198623

345)

= 11987734

sdot Δ3(119861

1198623

123) sdot Δ

3(119861

1198623

345)

= 11987734

sdot (1198611198623

123+ 119861

1198623

124) sdot (119861

1198623

356+ 119861

1198623

456)

= 11987734

sdot 1198611198623

123sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

123sdot 119861

1198623

456

+ 11987734

sdot 1198611198623

124sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

124sdot 119861

1198623

456

(35)



+ 2= 211987734 ∘ ( )Δ3



sharing a vertex








119899

119901=2cup119896

119902=0119881119901119902

biconnlet 119881119899119896

119883= 119866 isin 119881

119899119896


vention 11988110




1119901=

sum1198661015840isin119883119901119902 120590

11986610158401198611198661015840

1119901isin B

119901119902sub S(119881)


119901119902

sube

119881119901119902



1198661015840 gt 0 (ii)

sum1198661015840isinA 120590


define Ψ119899119896


Ψ10

= 1

Ψ1119896

= 0 if 119896 gt 0

(36)

Ψ119899119896

=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1)Φ119901119902

119894119894+1119894+119901minus1

sdotΔ119901minus1

119894(Ψ

119899minus119901+1119896minus119902

))

(37)

Then Ψ119899119896

= sum119866isin119881119899119896

119883

120572119866119878119866

1119899 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1|autC|



119881120583

119884= 119866 isin 119881

(119901minus1)120583+1119902120583


119866) = 120583

(38)





119901119902


119901119902

1119901=

sum1198661015840isin119884

12059011986610158401198611198661015840

1119901isin B

119901119902sub S(119881)



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin


1205950

= 1

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

(39)

Then 120595120583 = sum119866isin119881120583

119884

120572119866119878119866

1(119901minus1)120583+1with 120572





119866isinC 120572119866= 1|aut (C)|



(119901minus1)120583+1119902120583




120595120583

=

1

120583

times sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(119894


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(40)



1205951

= 120601119901119902


119901119902

1119901 (41)


120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


119894(120595

1198941


)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941


119894(120595

1198942


)

+ sdot sdot sdot +


sum


Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

(a)

1 2

(b)

3

21

(c)


12isin S(119881)


12sdot 119877

13sdot 119877

23isin S(119881)

otimes3


otimes119899

119877119894119895= 1

otimes119894minus1



(14)


119894119895isin S(119881)

otimes119899 and 119877119894119895

isin S(119881)otimes1198991015840

with 119899 = 1198991015840



(ii) a tensor 119877119894119895


a graph on 1 119899


and 119864(119866) = 119894119896 119895119896119896=1119898



S(119881)otimes119899



119894119896119895119896



1119899isin S(119881)


119894119896119895119896

119878119866

1119899=

119898

prod

119896=1

119877119894119896119895119896

(15)


1015840



1015840

119878119866

120590(1)120590(119899)=

119898

prod

119896=1

119877120590(119894119896)120590(119895119896) (16)



1

3 4

2

(a)

5

3

4

1

2

(b)






5 3 997891rarr 4 4 997891rarr 2


1015840



119896=0S(119881)





119899) ∙ (119904

1015840


1015840

1198991015840)


119899otimes 119904

1015840


1015840

1198991015840

(18)

where 119904119894 1199041015840




1otimes 119904

2997891rarr 119904

2otimes 119904

1


otimes119896minus1



1015840 we define ∙119894119895


rarr S(119881)otimes119899+1198991015840



119899) ∙119894119895(1199041015840


1015840

1198991015840)

= (120591119899minus1

∘ sdot sdot sdot ∘ 120591119894) (119904


119899)


119895minus1) (119904

1015840


1015840

1198991015840)



119899otimes 119904

119894

otimes 1199041015840

119895otimes 119904

1015840


1199041015840


1015840

1198991015840

(19)

where 119904119894(resp 1199041015840

119895) means that 119904

119894(resp 1199041015840



isin S(119881)otimes119899 and 119878119866

1015840

11198991015840 isin

S(119881)otimes1198991015840


119878119866

1119899∙1198941198951198781198661015840

11198991015840 = 119878

119866

120590(1)120590(119899)sdot 1198781198661015840

1205901015840(1)120590

1015840(1198991015840) (20)

where 119878119866120590(1)120590(119899)

1198781198661015840

1205901015840(1)120590

1015840(1198991015840)isin S(119881)

otimes119899+1198991015840

120590(119896) = 119896 if 1 le 119896 lt

119894 120590(119894) = 119899 120590(119896) = 119896 minus 1 if 119894 lt 119896 le 119899 and 1205901015840

(119896) = 119896 + 119899 + 1

if 1 le 119896 lt 119895 1205901015840(119895) = 119899 + 1 1205901015840(119896) = 119896 + 119899 if 119895 lt 119896 le 1198991015840


1

3 41

32 2119866prime119866

(a)

5

6

1

4 3

2

119867

(b)



and 1198781198661015840

123isin S(119881)


=

119878119866

1234321198781198661015840

123isin S(119881)

otimes6


∙1198941198951198781198661015840



otimes119894minus1


S(119881)otimes119899




119894119895= sdot

119899∘ ∙

119894119895 S(119881)

otimes119899

times S(119881)otimes1198991015840

997888rarr S(119881)otimes119899+1198991015840

minus1

(21)

The tensor 1198781198661119899

1198941198951198781198661015840


1015840

minus 1


6 Linear Mappings


119899119896sub S(119881)


119899119896

biconn Let B =

B10⨁B

20⨁

infin

119899=3⨁

119896max(119899)

119896=1B

119899119896sub T(S(119881)) where 119896max(119899) =



Δ (1) = 1 otimes 1

Δ (119861119866

1119899) =

1

119899

119899

sum

119894=1

Δ119894(119861

119866

1119899) if 119899 gt 1

(22)


119866

1119899isin S(119881)



(i) 120590119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894

119895 + 1 if 119894 + 1 le 119895 le 119899

(ii) 120584119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894 minus 1

119895 + 1 if 119894 le 119895 le 119899

(23)


otimes119899


equation

Δ119894(119861

119866

1119899) = 119861

119866

120590119894(1)120590

119894(119899)

+ 119861119866

120584119894(1)120584

119894(119899)

= 119861119866

1119894+1119894+2119899+1+ 119861

119866

1119894119894+1119899+1

(24)


1119894119899+1(resp

119861119866

1119894+1119894+2119899+1





isin S(119881)otimes1198991015840


1015840


Δ119894(119861

119866

120587(1)120587(119899)) = 119861

119866

120584119894(120587(1))120584

119894(120587(119899))

(25)



Δ119894∘ Δ

119894= Δ

119894+1∘ Δ

119894 (26)


S(119881)otimes119899

rarr


and Δ119894 S(119881)

otimes119899+1



S(119881)otimes119899+1



119894 Δ119898

119894

S(119881)otimes119899


Δ1

119894= Δ

119894 (27)

Δ119898

119894= Δ

119894∘ Δ

119898minus1

119894 (28)

where Δ119894

S(119881)otimes119899+119898minus1






S(119881)otimes119899+119898minus1





= ⨁infin

120583=0B120583 where B0



otimes119899 and 119861119901sub S(119881)

otimes119901 by119860119899119894119895119861119901



119899 119887 isin 119861

119901) Also let B

2=

B20

and B119899= ⨁

119896max(119899)

119896=1B



119899

120583 as follows

Blowast

119899

120583

= ⋃


120583=119899+120583minus1

1198991

⋃

1198941=1

sdot sdot sdot

119899120583minus1

⋃

119894120583minus1=1

119899120583+sdotsdotsdot+119899

2minus120583+2

⋃

1198951=1

sdot sdot sdot

119899120583

⋃

119895120583minus1=1

B1198991

11989411198951

(B1198992

11989421198952


119894120583minus2119895120583minus2

(B119899120583minus1

119894120583minus1119895120583minus1

B119899120583

)) sdot sdot sdot)

(29)

Also define

B1015840

119899

lowast120583

= ⋃

120587isin119878119899

119878119866

120587(1)120587(119899)| 119878

119866

1119899isin B

lowast

119899

120583

(30)


and 119878119866


define

B120583

=

infin

⨁

119899=120583+1

B1015840

119899

lowast120583

(31)


otimes119899




prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886) (32)


119886(119899119886)isin S(119881)

otimes119899 119866119886is a


119861119866119886

1119899119886

isin S(119881)otimes119899119886 and 120590

119886 1 119899

119886 rarr 119883

119886sube 1 119899 is a


119899



Δ119894(

120583

prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886)) =

120583

prod

119886=1

Δ119894(119861

119866119886

120590119886(1)120590

119886(119899119886)) (33)










119866

1119899isin


119894119894+1119894+119899minus1= 119861

119866

120587119894(1)120587

119894(119899)




119894119894+1119894+119899minus1

with Δ119899minus1

119894

119861119866

119894119894+1119894+119899minus1∘ Δ

119899minus1

119894 S(119881)

otimes119901


(34)



∘ Δ119899minus1

119894produce







∘ Δ



120587119894(1)120587

119894(119899)

∘ Δ119894


123sdot1198611198623

345isin S(119881)


123= 119877

12sdot 119877

23sdot 119877

13isin

S(119881)otimes3 Let 119879


1198792

12=

11987712


34∘ Δ

3to119867 yields

11987734

∘ Δ3(119861

1198623

123sdot 119861

1198623

345)

= 11987734

sdot Δ3(119861

1198623

123) sdot Δ

3(119861

1198623

345)

= 11987734

sdot (1198611198623

123+ 119861

1198623

124) sdot (119861

1198623

356+ 119861

1198623

456)

= 11987734

sdot 1198611198623

123sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

123sdot 119861

1198623

456

+ 11987734

sdot 1198611198623

124sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

124sdot 119861

1198623

456

(35)



+ 2= 211987734 ∘ ( )Δ3



sharing a vertex








119899

119901=2cup119896

119902=0119881119901119902

biconnlet 119881119899119896

119883= 119866 isin 119881

119899119896


vention 11988110




1119901=

sum1198661015840isin119883119901119902 120590

11986610158401198611198661015840

1119901isin B

119901119902sub S(119881)


119901119902

sube

119881119901119902



1198661015840 gt 0 (ii)

sum1198661015840isinA 120590


define Ψ119899119896


Ψ10

= 1

Ψ1119896

= 0 if 119896 gt 0

(36)

Ψ119899119896

=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1)Φ119901119902

119894119894+1119894+119901minus1

sdotΔ119901minus1

119894(Ψ

119899minus119901+1119896minus119902

))

(37)

Then Ψ119899119896

= sum119866isin119881119899119896

119883

120572119866119878119866

1119899 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1|autC|



119881120583

119884= 119866 isin 119881

(119901minus1)120583+1119902120583


119866) = 120583

(38)





119901119902


119901119902

1119901=

sum1198661015840isin119884

12059011986610158401198611198661015840

1119901isin B

119901119902sub S(119881)



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin


1205950

= 1

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

(39)

Then 120595120583 = sum119866isin119881120583

119884

120572119866119878119866

1(119901minus1)120583+1with 120572





119866isinC 120572119866= 1|aut (C)|



(119901minus1)120583+1119902120583




120595120583

=

1

120583

times sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(119894


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(40)



1205951

= 120601119901119902


119901119902

1119901 (41)


120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


119894(120595

1198941


)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941


119894(120595

1198942


)

+ sdot sdot sdot +


sum


Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

3 41

32 2119866prime119866

(a)

5

6

1

4 3

2

119867

(b)



and 1198781198661015840

123isin S(119881)


=

119878119866

1234321198781198661015840

123isin S(119881)

otimes6


∙1198941198951198781198661015840



otimes119894minus1


S(119881)otimes119899




119894119895= sdot

119899∘ ∙

119894119895 S(119881)

otimes119899

times S(119881)otimes1198991015840

997888rarr S(119881)otimes119899+1198991015840

minus1

(21)

The tensor 1198781198661119899

1198941198951198781198661015840


1015840

minus 1


6 Linear Mappings


119899119896sub S(119881)


119899119896

biconn Let B =

B10⨁B

20⨁

infin

119899=3⨁

119896max(119899)

119896=1B

119899119896sub T(S(119881)) where 119896max(119899) =



Δ (1) = 1 otimes 1

Δ (119861119866

1119899) =

1

119899

119899

sum

119894=1

Δ119894(119861

119866

1119899) if 119899 gt 1

(22)


119866

1119899isin S(119881)



(i) 120590119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894

119895 + 1 if 119894 + 1 le 119895 le 119899

(ii) 120584119894 119895 997891997888rarr

119895 if 1 le 119895 le 119894 minus 1

119895 + 1 if 119894 le 119895 le 119899

(23)


otimes119899


equation

Δ119894(119861

119866

1119899) = 119861

119866

120590119894(1)120590

119894(119899)

+ 119861119866

120584119894(1)120584

119894(119899)

= 119861119866

1119894+1119894+2119899+1+ 119861

119866

1119894119894+1119899+1

(24)


1119894119899+1(resp

119861119866

1119894+1119894+2119899+1





isin S(119881)otimes1198991015840


1015840


Δ119894(119861

119866

120587(1)120587(119899)) = 119861

119866

120584119894(120587(1))120584

119894(120587(119899))

(25)



Δ119894∘ Δ

119894= Δ

119894+1∘ Δ

119894 (26)


S(119881)otimes119899

rarr


and Δ119894 S(119881)

otimes119899+1



S(119881)otimes119899+1



119894 Δ119898

119894

S(119881)otimes119899


Δ1

119894= Δ

119894 (27)

Δ119898

119894= Δ

119894∘ Δ

119898minus1

119894 (28)

where Δ119894

S(119881)otimes119899+119898minus1






S(119881)otimes119899+119898minus1





= ⨁infin

120583=0B120583 where B0



otimes119899 and 119861119901sub S(119881)

otimes119901 by119860119899119894119895119861119901



119899 119887 isin 119861

119901) Also let B

2=

B20

and B119899= ⨁

119896max(119899)

119896=1B



119899

120583 as follows

Blowast

119899

120583

= ⋃


120583=119899+120583minus1

1198991

⋃

1198941=1

sdot sdot sdot

119899120583minus1

⋃

119894120583minus1=1

119899120583+sdotsdotsdot+119899

2minus120583+2

⋃

1198951=1

sdot sdot sdot

119899120583

⋃

119895120583minus1=1

B1198991

11989411198951

(B1198992

11989421198952


119894120583minus2119895120583minus2

(B119899120583minus1

119894120583minus1119895120583minus1

B119899120583

)) sdot sdot sdot)

(29)

Also define

B1015840

119899

lowast120583

= ⋃

120587isin119878119899

119878119866

120587(1)120587(119899)| 119878

119866

1119899isin B

lowast

119899

120583

(30)


and 119878119866


define

B120583

=

infin

⨁

119899=120583+1

B1015840

119899

lowast120583

(31)


otimes119899




prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886) (32)


119886(119899119886)isin S(119881)

otimes119899 119866119886is a


119861119866119886

1119899119886

isin S(119881)otimes119899119886 and 120590

119886 1 119899

119886 rarr 119883

119886sube 1 119899 is a


119899



Δ119894(

120583

prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886)) =

120583

prod

119886=1

Δ119894(119861

119866119886

120590119886(1)120590

119886(119899119886)) (33)










119866

1119899isin


119894119894+1119894+119899minus1= 119861

119866

120587119894(1)120587

119894(119899)




119894119894+1119894+119899minus1

with Δ119899minus1

119894

119861119866

119894119894+1119894+119899minus1∘ Δ

119899minus1

119894 S(119881)

otimes119901


(34)



∘ Δ119899minus1

119894produce







∘ Δ



120587119894(1)120587

119894(119899)

∘ Δ119894


123sdot1198611198623

345isin S(119881)


123= 119877

12sdot 119877

23sdot 119877

13isin

S(119881)otimes3 Let 119879


1198792

12=

11987712


34∘ Δ

3to119867 yields

11987734

∘ Δ3(119861

1198623

123sdot 119861

1198623

345)

= 11987734

sdot Δ3(119861

1198623

123) sdot Δ

3(119861

1198623

345)

= 11987734

sdot (1198611198623

123+ 119861

1198623

124) sdot (119861

1198623

356+ 119861

1198623

456)

= 11987734

sdot 1198611198623

123sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

123sdot 119861

1198623

456

+ 11987734

sdot 1198611198623

124sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

124sdot 119861

1198623

456

(35)



+ 2= 211987734 ∘ ( )Δ3



sharing a vertex








119899

119901=2cup119896

119902=0119881119901119902

biconnlet 119881119899119896

119883= 119866 isin 119881

119899119896


vention 11988110




1119901=

sum1198661015840isin119883119901119902 120590

11986610158401198611198661015840

1119901isin B

119901119902sub S(119881)


119901119902

sube

119881119901119902



1198661015840 gt 0 (ii)

sum1198661015840isinA 120590


define Ψ119899119896


Ψ10

= 1

Ψ1119896

= 0 if 119896 gt 0

(36)

Ψ119899119896

=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1)Φ119901119902

119894119894+1119894+119901minus1

sdotΔ119901minus1

119894(Ψ

119899minus119901+1119896minus119902

))

(37)

Then Ψ119899119896

= sum119866isin119881119899119896

119883

120572119866119878119866

1119899 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1|autC|



119881120583

119884= 119866 isin 119881

(119901minus1)120583+1119902120583


119866) = 120583

(38)





119901119902


119901119902

1119901=

sum1198661015840isin119884

12059011986610158401198611198661015840

1119901isin B

119901119902sub S(119881)



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin


1205950

= 1

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

(39)

Then 120595120583 = sum119866isin119881120583

119884

120572119866119878119866

1(119901minus1)120583+1with 120572





119866isinC 120572119866= 1|aut (C)|



(119901minus1)120583+1119902120583




120595120583

=

1

120583

times sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(119894


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(40)



1205951

= 120601119901119902


119901119902

1119901 (41)


120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


119894(120595

1198941


)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941


119894(120595

1198942


)

+ sdot sdot sdot +


sum


Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












S(119881)otimes119899+119898minus1





= ⨁infin

120583=0B120583 where B0



otimes119899 and 119861119901sub S(119881)

otimes119901 by119860119899119894119895119861119901



119899 119887 isin 119861

119901) Also let B

2=

B20

and B119899= ⨁

119896max(119899)

119896=1B



119899

120583 as follows

Blowast

119899

120583

= ⋃


120583=119899+120583minus1

1198991

⋃

1198941=1

sdot sdot sdot

119899120583minus1

⋃

119894120583minus1=1

119899120583+sdotsdotsdot+119899

2minus120583+2

⋃

1198951=1

sdot sdot sdot

119899120583

⋃

119895120583minus1=1

B1198991

11989411198951

(B1198992

11989421198952


119894120583minus2119895120583minus2

(B119899120583minus1

119894120583minus1119895120583minus1

B119899120583

)) sdot sdot sdot)

(29)

Also define

B1015840

119899

lowast120583

= ⋃

120587isin119878119899

119878119866

120587(1)120587(119899)| 119878

119866

1119899isin B

lowast

119899

120583

(30)


and 119878119866


define

B120583

=

infin

⨁

119899=120583+1

B1015840

119899

lowast120583

(31)


otimes119899




prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886) (32)


119886(119899119886)isin S(119881)

otimes119899 119866119886is a


119861119866119886

1119899119886

isin S(119881)otimes119899119886 and 120590

119886 1 119899

119886 rarr 119883

119886sube 1 119899 is a


119899



Δ119894(

120583

prod

119886=1

119861119866119886

120590119886(1)120590

119886(119899119886)) =

120583

prod

119886=1

Δ119894(119861

119866119886

120590119886(1)120590

119886(119899119886)) (33)










119866

1119899isin


119894119894+1119894+119899minus1= 119861

119866

120587119894(1)120587

119894(119899)




119894119894+1119894+119899minus1

with Δ119899minus1

119894

119861119866

119894119894+1119894+119899minus1∘ Δ

119899minus1

119894 S(119881)

otimes119901


(34)



∘ Δ119899minus1

119894produce







∘ Δ



120587119894(1)120587

119894(119899)

∘ Δ119894


123sdot1198611198623

345isin S(119881)


123= 119877

12sdot 119877

23sdot 119877

13isin

S(119881)otimes3 Let 119879


1198792

12=

11987712


34∘ Δ

3to119867 yields

11987734

∘ Δ3(119861

1198623

123sdot 119861

1198623

345)

= 11987734

sdot Δ3(119861

1198623

123) sdot Δ

3(119861

1198623

345)

= 11987734

sdot (1198611198623

123+ 119861

1198623

124) sdot (119861

1198623

356+ 119861

1198623

456)

= 11987734

sdot 1198611198623

123sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

123sdot 119861

1198623

456

+ 11987734

sdot 1198611198623

124sdot 119861

1198623

356+ 119877

34sdot 119861

1198623

124sdot 119861

1198623

456

(35)



+ 2= 211987734 ∘ ( )Δ3



sharing a vertex








119899

119901=2cup119896

119902=0119881119901119902

biconnlet 119881119899119896

119883= 119866 isin 119881

119899119896


vention 11988110




1119901=

sum1198661015840isin119883119901119902 120590

11986610158401198611198661015840

1119901isin B

119901119902sub S(119881)


119901119902

sube

119881119901119902



1198661015840 gt 0 (ii)

sum1198661015840isinA 120590


define Ψ119899119896


Ψ10

= 1

Ψ1119896

= 0 if 119896 gt 0

(36)

Ψ119899119896

=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1)Φ119901119902

119894119894+1119894+119901minus1

sdotΔ119901minus1

119894(Ψ

119899minus119901+1119896minus119902

))

(37)

Then Ψ119899119896

= sum119866isin119881119899119896

119883

120572119866119878119866

1119899 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1|autC|



119881120583

119884= 119866 isin 119881

(119901minus1)120583+1119902120583


119866) = 120583

(38)





119901119902


119901119902

1119901=

sum1198661015840isin119884

12059011986610158401198611198661015840

1119901isin B

119901119902sub S(119881)



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin


1205950

= 1

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

(39)

Then 120595120583 = sum119866isin119881120583

119884

120572119866119878119866

1(119901minus1)120583+1with 120572





119866isinC 120572119866= 1|aut (C)|



(119901minus1)120583+1119902120583




120595120583

=

1

120583

times sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(119894


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(40)



1205951

= 120601119901119902


119901119902

1119901 (41)


120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


119894(120595

1198941


)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941


119894(120595

1198942


)

+ sdot sdot sdot +


sum


Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 2= 211987734 ∘ ( )Δ3



sharing a vertex








119899

119901=2cup119896

119902=0119881119901119902

biconnlet 119881119899119896

119883= 119866 isin 119881

119899119896


vention 11988110




1119901=

sum1198661015840isin119883119901119902 120590

11986610158401198611198661015840

1119901isin B

119901119902sub S(119881)


119901119902

sube

119881119901119902



1198661015840 gt 0 (ii)

sum1198661015840isinA 120590


define Ψ119899119896


Ψ10

= 1

Ψ1119896

= 0 if 119896 gt 0

(36)

Ψ119899119896

=

1

119896 + 119899 minus 1

times

119896

sum

119902=0

119899

sum

119901=2

119899minus119901+1

sum

119894=1

((119902 + 119901 minus 1)Φ119901119902

119894119894+1119894+119901minus1

sdotΔ119901minus1

119894(Ψ

119899minus119901+1119896minus119902

))

(37)

Then Ψ119899119896

= sum119866isin119881119899119896

119883

120572119866119878119866

1119899 where 120572

119866isin Q and 119883 =

cup119899

119901=2cup119896


E(119881119899119896119883


119866isinC 120572119866= 1|autC|



119881120583

119884= 119866 isin 119881

(119901minus1)120583+1119902120583


119866) = 120583

(38)





119901119902


119901119902

1119901=

sum1198661015840isin119884

12059011986610158401198611198661015840

1119901isin B

119901119902sub S(119881)



1198661015840 gt 0 (ii) sum

1198661015840isinA 120590

1198661015840 =


isin


1205950

= 1

120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

(39)

Then 120595120583 = sum119866isin119881120583

119884

120572119866119878119866

1(119901minus1)120583+1with 120572





119866isinC 120572119866= 1|aut (C)|



(119901minus1)120583+1119902120583




120595120583

=

1

120583

times sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(119894


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(40)



1205951

= 120601119901119902


119901119902

1119901 (41)


120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


119894(120595

1198941


)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941


119894(120595

1198942


)

+ sdot sdot sdot +


sum


Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120595120583

=

1

120583

times sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum

119886119901=(119901minus1)(119894


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(40)



1205951

= 120601119901119902


119901119902

1119901 (41)


120595120583

=

1

120583

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

120583minus1

)

=

1

120583 (120583 minus 1)

(119901minus1)(120583minus1)+1

sum

119894=1

120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894

( sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


))

=

1

120583 (120583 minus 1)

times sum


(119901minus1)(120583minus1)+1

sum

119894=1

(Δ119901minus1

119894(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot120601119901119902

119894119894+1119894+119901minus1sdot Δ

119901minus1

119894(120595

1198941


))

=

1

120583 (120583 minus 1)

times ( sum


(

(119901minus1)1198941+1

sum

119894=1

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum



sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


119894(120595

1198941


)

+

(119901minus1)(1198941+1198942)+2

sum

119894=(119901minus1)1198941+2

Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


)

sdot 120601119901119902

119894119894+1119894+119901minus1

sdot (1205951198941


119894(120595

1198942


)

+ sdot sdot sdot +


sum


Δ119901minus1

119894

(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1


sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










sdot (1205951198941


119894(120595

119894119901

))))

=

1

120583 (120583 minus 1)

times ( sum


((1198941+ 1)

times (

(119901minus1)1198941+119901

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+119901+1

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941+1


) + (1198942+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+119901+1

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941

otimes 1205951198942+1


)

+ sdot sdot sdot + (119894119901+ 1)

times (

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot 120601119901119902

119894119894+1119894+119901minus1sdot (120595

1198941


)))

=

1

120583 (120583 minus 1)

times (

120583minus1

sum

1198941=0

sum


1

1198941+

120583minus1

sum

1198942=0

sum


2

1198942

+ sdot sdot sdot +

120583minus1

sum

119894119901=0

sum


119901

119894119901)

times(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

=

1

120583

sum


(

(119901minus1)1198941+1

sum

1198861=1

(119901minus1)(1198941+1198942)+2

sum

1198862=(119901minus1)1198941+2

sdot sdot sdot


sum


120601119901119902

1198861119886119901

)

sdot (1205951198941


)

(42)


120583

119884is



Let 119884 sube 119881119901119902




120583




119901119902


sum

CisinE(119881120583

119884)

1

|autC|

=

((119901 minus 1) 120583 + 1)120583minus2

120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(43)


119899minus1

sum

119894=1

(

119899

119894) 119894

119894minus1


= 2 (119899 minus 1) 119899119899minus2

(44)

119899

sum

119894=0

(

119899

119894) (119909 + 119894)

119894minus1


= (

1

119909

+

1

119910

) (119909 + 119910 + 119899)119899minus1

(45)




119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901 the iden-

tity

sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(46)


sum

1198941+sdotsdotsdot+119894119901=119899

(

119899

1198941 119894

119901

) (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

119899

sum

1198941=0

sum


(

119899

1198941

)(

119899 minus 1198941

1198942 119894

119901

)

times (1199091+ 119894

1)1198941minus1

sdot sdot sdot (119909119901+ 119894

119901)

119894119901minus1

=

1199092+ sdot sdot sdot + 119909

119901


119901

times

119899

sum

1198941=0

(

119899

1198941

) (1199091+ 119894

1)1198941minus1

(1199092+ sdot sdot sdot + 119909

119901+ 119899 minus 119894

1)


=

1199092+ sdot sdot sdot + 119909

119901


119901

times (

1

1199091

+

1

1199092+ sdot sdot sdot + 119909

119901

)(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

=

1199091+ sdot sdot sdot + 119909

119901


119901

(1199091+ sdot sdot sdot + 119909

119901+ 119899)

119899minus1

(47)


119884)(1


119868 (0) = 1

119868 (120583) =

1

120583

sum

AisinE(119884)

1

|autA|

times sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901)

(48)


1)120583minus2


0 1

119868 (0) = 1

119868 (1) = sum

AisinE(119884)

1

|autA|

(49)


119868 (120583) =

1

120583

sum


((119901 minus 1) 1198941+ 1)


1) sdot sdot sdot 119868 (119894

119901) sum

AisinE(119884)

1

|autA|

(50)

=

119901120583minus1

120583

sum


1


119901

((119901 minus 1) 1198941+ 1)

1198941minus1


119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(51)

=

119901120583minus1

120583


times sum


(

120583 minus 1

1198941 119894

119901

)(1198941+

1

119901 minus 1

)

1198941minus1


1

119901 minus 1

)

119894119901minus1

( sum

AisinE(119884)

1

|autA|

)

120583

(52)

=

1

120583


(119901 minus 1)119901minus1


119901

119901 minus 1

)

120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583 (53)

=

1

120583

((119901 minus 1) 120583 + 1)120583minus2

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(54)




119881119901119902


120583

119884| 119881(119866) =



120583

(119901 sum

AisinE(119884)

1

|autA|

)

120583

(55)


120583

119884)


(120583 + 1)120583minus1

= 119899119899minus2

(56)



119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119869 (1) = 1

119869 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

119894 (119899 minus 119894) 119869 (119894) 119869 (119899 minus 119894)

(57)



119879 (1) = 1

119879 (119899) =

1

2 (119899 minus 1)

119899minus1

sum

119894=1

(

119899

119894) 119894 (119899 minus 119894) 119879 (119894) 119879 (119899 minus 119894)

(58)



(119901 minus 1) 120583

120583

(59)



2120583120583

(60)

Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article An Algebraic Representation of Graphs and...

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