GAO YAN-YAN,TANG GAO-HUAAND CHEN JIAN-LONG

(1.Department of Mathematics,Southeast University,Nanjing,210096)

(2.School of Mathematical Sciences,Guangxi Education University,Nanning,530001)

On Commuting Graph of Group RingZnS3∗

GAO YAN-YAN1,TANG GAO-HUA2AND CHEN JIAN-LONG1

(1.Department of Mathematics,Southeast University,Nanjing,210096)

(2.School of Mathematical Sciences,Guangxi Education University,Nanning,530001)

The commuting graph of an arbitrary ringR,denoted byΓ(R),is a graph whose vertices are all non-central elements ofR,and two distinct verticesaandbare adjacent if and only ifab=ba.In this paper,we investigate the connectivity and the diameter ofΓ(ZnS3).We show thatΓ(ZnS3)is connected if and only ifnis not a prime number.IfΓ(ZnS3)is connected then diam(Γ(ZnS3))=3,while ifΓ(ZnS3)is disconnected then every connected component ofΓ(ZnS3)must be a complete graph with same size,and we completely determine the vertice set of every connected component.

group ring,commuting graph,connected component,diameter of a graph

1 Introduction

LetGbe a group andRa ring.We denote byRGthe set of all formal linear combinations of the form

whereag∈Randag=0 almost everywhere,that is,only a finite number of coefficients are different from 0 in each of these sums.Notice that it follows from our de fi nition that given two elements

The commuting graph of an arbitrary ringRdenoted byΓ(R)is a graph with vertex setV(R)=R(R),whereZ(R)is the center ofR,and two distinct verticesaandbare adjacent if and only ifab=ba.The notion of commuting graph of a ring was first introduced by Akbariet al.[1]in 2004.They investigated some properties ofΓ(R),wheneverRis a finite semisimple ring.For any finite fieldF,they obtained connectivity,minimum degree, maximum degree and clique number ofΓ(Mn(F)).Also it was shown that for any two finite semisimple ringsRandS,ifΓ(R)(S),then there are commutative semisimple ringsR1andS1and semisimple ringTsuch that

The commuting graphs of some special rings have also been studied(see[2–4]).

Group rings are very interesting algebraic structure.For a group ringZnS3,the properties of commuting graph can re fl ect its some structures.In this paper,we investigate some properties ofΓ(ZnS3),where

is the symmetric group of order 6,and

is the modulenresidue class ring.Given a group ringRGand a finite subsetXof the groupG,we denote bythe following element ofRG:

In addition,the distinct conjugacy classes ofS3are

In this paper,all graphs are simple and undirected and|G|denotes the number of vertices of the graphG.We writex∈V(G)whenxis a vertex ofG.A path of lengthrfrom a vertexxto another vertexyinGis a sequence ofr+1 distinct vertices starting withxand ending withysuch that consecutive vertices are adjacent.For a connected graphH, the diameter ofHis denoted by diam(H).An induced subgraph ofGthat is maximal and connected,is called a connected component ofG.

In this paper,we investigate the connectivity and the diameter ofΓ(ZnS3).We show thatΓ(ZnS3)is connected if and only ifnis not a prime number.IfΓ(ZnS3)is connected then diam(Γ(ZnS3))=3,while ifΓ(ZnS3)is disconnected then every connected component ofΓ(ZnS3)must be a complete graph with same size,and we completely determine the vertice set of every connected component.

LetRbe a ring andR∗=R{0}.The ring ofnbynfull matrices over a ringRis denoted byMn(R).is a quadratic extension of the fieldZp.

2 Main Results

Lemma 2.1([1],Theorem 2)If F is a finite field,then Γ(M2(F))is a graph with|F|2+|F|+1connected components of size|F|2−|F|,each of which is a complete graph.

Lemma 2.2Let n be an arbitrary positive integer.Then

By Lemma 2.2,we know thatγ∈Z(ZnS3).Hence,there does not exist a vertexγofV(ZnS3)such thata—γ—bis a path ofΓ(ZnS3).Hence,

The proof is completed.

Lemma 2.3([5],Theorem 2.6.8)A ring R is semisimple if and only if it is a direct sum of matrix algebras over division rings:

Ifα3andβ3are not in the same connected component ofM2(Zp),then there is no edge betweenαandβ.By Lemma 2.1,we know thatΓ(M2(Zp))is a graph withp2+p+1 connected components of sizep2−p,each of which is a complete graph.Hence,Γ(ZpS3)is a graph withp2+p+1 connected components of sizep4−p3,each of which is a complete graph.This completes the proof.

Theorem 2.3Γ(Z2S3)is a graph with7connected components of size8,each of which is a complete graph.

Moreover,we can conclude that each connected componentAi(i=1,2,···,7)is a complete graph.This completes our proof.

Theorem 2.4Γ(Z3S3)is a graph with13connected components of size54,each of which is a complete graph.And the following13sets are all the sets of vertices of the connected components of Γ(Z3S3):

Proof.By straightforward computation we derive that the number of elements of each set in the theorem above is 54,and each vertex ofΓ(Z3S3)must belong to and only belong to oneAk(1≤k≤13).Moreover,it is easy to verify that for any

This completes our proof.

Now we consider the commuting graphs ofZnS3whennhas at least two distinct prime divisors.In order to get our results,we need the following lemma.

Lemma 2.6([6],Proportion 8.1.20)Let R be a commutative noetherian ring and let G be an arbitrary group.Then there exist finitely many indecomposable rings R1,R2,···,Rn such that

Theorem 2.5Let p be a prime number.Then Γ(Z2pS3)is a connected graph anddiam(Γ(Z2pS3))=3.

Proof.(1)Ifp=2,by Theorem 2.1,the result follows.

(2)Ifp=3,by Lemma 2.6,we have

LetA1,A2,···,A7be the sets of vertices of the connected components ofΓ(Z2S3) and let the sets of vertices of the connected components ofΓ(Z3S3)areB1,B2,···,B13. Moreover,

By symmetry,we only consider the following cases:

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Case 1.Forα=(α1,α2)∈Z6S3,β=(β1,β2)∈Z6S3,whereα1∈V(Z2S3),α2∈Z(Z2S3),β1∈Z(Z3S3),β2∈V(Z3S3),thenα—βis an edge ofΓ(Z6S3).

Case 2.Forα=(α1,α2)∈Z6S3,β=(β1,β2)∈Z6S3,whereα1,β1∈Z(Z2S3),α2,β2∈V(Z3S3),ifα2,β2∈Bj,thenα—βis an edge ofΓ(Z6S3),otherwise,(α1,α2)—(0)—(β1,0)—(β1,β2)is a path ofΓ(Z6S3),whereα2.

Case 3.Forα=(α1,α2)∈Z6S3,β=(β1,β2)∈Z6S3,whereα1,β1∈V(Z2S3),β2∈V(Z3S3),α2∈Z(Z2S3),ifα1,β1∈Ai,thenα—βis an edge ofΓ(Z6S3),otherwise, (α1,0)—(0)—(β1,β2)is a path ofΓ(Z6S3),whereβ2.

Case 4.Letα=(α1,α2)∈Z6S3,β=(β1,β2)∈Z6S3,whereα1,β1∈V(Z2S3),α2,β2∈V(Z3S3).

Subcase 4.1.Ifα1,β1∈Ai,α2,β2∈Bj,thenα—βis an edge ofΓ(Z6S3).

Therefore,we can conclude thatΓ(Z6S3)is a connected graph and diam(Γ(Z6S3))=3.

(3)Ifp＞3,by Lemmas 2.5 and 2.6,we have

Then,by symmetry,for anyα=(α1,α2,α3,α4)∈Z2pS3and anyβ=(β1,β2,β3,βn)∈Z2pS3,whereα1,β1∈Z2S3,α2,β2,α3,β3∈Zp,α4,β4∈M2(Zp),we have the following cases to consider.

First,letA1,A2,···,A7be the sets of vertices of the connected components ofΓ(Z2S3). By Lemma 2.1,we know that there arep2+p+1 connected components inΓ((M2(Zp)) and we denote them asCi,i=1,2,···,p2+p+1.

Case 2.Assume thatα1,β1∈Z(Z2S3),α4,β4∈V(M2(Zp)).Ifα4,β4∈Cifor somei,thenα—βis an edge ofΓ(ZnS3).Otherwise,(α1,α2,α3,α4)—(0,0,0)—(β1,0,0,0)—(β1,β2,β3,β4)is a path ofΓ(Z2pS3),whereα4.

Case 3.Assume thatα1∈Z(Z2S3),β1∈V(Z2S3),α4,β4∈V(M2(Zp)).By similar argument above,we have the same results.

Case 4.Assume thatα1,β1∈V(Z2S3),α4,β4∈V(M2(Zp)).

Subcase 4.1.Suppose thatα1,β1∈Ai,α4,β4∈Cj,for somei,j.Thenα—βis an edge ofΓ(Z2pS3).

Therefore,Γ(Z2pS3)is a connected graph and diam(Γ(Z2pS3))=3.The proof is completed.

Theorem 2.6If n(＞1)is not a prime number,then Γ(ZnS3)is a connected graph anddiam(Γ(ZnS3))=3.

Notice that for anyα=(α1,α2,···,αm)∈R,α∈Z(R)if and only ifαi∈Z(Ri),i= 1,2,···,m.So for anyα=(α1,α2,···,αm)∈V(R)and anyβ=(β1,β2,···,βm)∈V(R), we consider the following three cases.

Case 1.Assume thatαi∈Z(Ri)orβi∈Z(Ri),i=1,2,···,m.Thenα—βis an edge ofΓ(R).

Case 2.Assume that there existsi∈{1,2,···,m}such thatαi∈Z(Ri)orβi∈Z(Ri). Without lose of generality,we can assume thatαi∈Z(Ri),and takeγi∈V(Ri)such thatβiγi=γiβi,whereγi/=βi.Setγ=(0,0,···,γi,0,···,0)∈R.Thenγ∈Z(R)andγ/=α,β.Soα—γ—βis an path ofΓ(R).

Consequently,Γ(R)must be connected and diam(Γ(R))≤3.Furthermore,noticing that there must exist an odd prime numberqsuch thatq/=pi,for anyi=1,2,···,m,we haveqa,qb∈V(R).Then by an argument similar to that of Theorem 2.1,we can conclude that there does not exist a vertexαofV(R)such thatqa—α—qbis a path ofΓ(R).Thus diam(Γ(R))=3.This completes our proof.

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[2]Abdollahi A.Commuting graphs of full matrix rings over finite fields.Linear Algebra Appl., 2008,428:2947–2954.

[3]Akbari S,Mohammadian A,Radjavi H,Raja P.On the diameters of commuting graphs.Linear Algebra Appl.,2006,418:161–176.

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Communicated by Du Xian-kun

16S34,20C05,05C12,05C40

A

1674-5647(2012)04-0313-11

date:Sept.30,2010.

The NSF(10971024)of China,the Specialized Research Fund(200802860024)for the Doctoral Program of Higher Education and the NSF(BK2010393)of Jiangsu Province.