The large sum graph related to comultiplication modules
aa r X i v : . [ m a t h . A C ] S e p THE LARGE SUM GRAPH RELATED TOCOMULTIPLICATION MODULES
H. ANSARI-TOROGHY AND F. MAHBOOBI-ABKENAR Abstract.
Let R be a commutative ring and M be an R -module.We define the large sum graph, denoted by ´ G ( M ), as a graphwith the vertex set of non-large submodules of M and two distinctvertices are adjacent if and only if N + K is a non-large submoduleof M . In this article, we investigate the connection between thegraph-theoretic properties of ´ G ( M ) and some algebraic propertiesof M when M is a comultiplication R -module. Introduction
Throughout this paper, R will denote a commutative ring with iden-tity and Z will denote the ring of integers.Let M be an R -module. We denote the set of all minimal submodulesof M by M in ( M ) and the sum of all minimal submodules of M by Soc ( M ). A submodule N of M is called large in M and denoted by N ✂ M ) in case for every submodule L of M , N ∩ L = 0. A module M is called a uniform module if the intersection of any two non-zerosubmodules of M is non-zero.A graph G is defined as the pair ( V ( G ) , E ( G )), where V ( G ) is theset of vertices of G and E ( G ) is the set of edges of G . For two distinctvertices a and b denoted by a − b means that a and b are adjacent.The degree of a vertex a of graph G which denoted by deg ( a ) is thenumber of edges incident on a . A regular graph is r -regular (or regularof degree r) if the degree of each vertex is r . If | V ( G ) | >
2, a path from a to b is a series of adjacent vertices a − v − v − ... − v n − b . In agraph G , the distance between two distinct vertices a and b , denoted Key words and phrases.
Graph, Non-large submodule, Comultiplication module.2010
Mathematics Subject Classification . Primary: 05C75. Secondary: 13A99,05C99. by d ( a, b ) is the length of the shortest path connecting a and b . If thereis not a path between a and b , d ( a, b ) = ∞ . The diameter of a graph G is diam ( G ) = sup { d ( a, b ) | a, b ∈ V ( G ) } . A graph G is called connected if for any vertices a and b of G there is a path between a and b . If not, G is disconnected . The girth of G , denoted by g ( G ), is the length of theshortest cycle in G . If G has no cycle, we define the girth of G to beinfinite. An r -partite graph is one whose vertex set can be partitionedinto r subsets such that no edge has both ends in any one subset. A complete r -partite graph is one each vertex is jointed to every vertexthat is not in the same subset. The complete bipartite (i.e, 2-partite)graph with part sizes m and n is denoted by K m,n . A star graph is acompleted bipartite graph with part sizes m = 1 or n = 1. A clique ofa graph is its maximal complete subgraph and the number of verticesin the largest clique of a graph G , denoted by ω ( G ), is called the cliquenumber of G . For a graph G = ( V, E ), a set S ⊆ V is an independent ifno two vertices in S are adjacent. The independence number α ( G ) is themaximum size of an independent set in G . The (open) neighbourhood N ( a ) of a vertex a ∈ V is the set of vertices which are adjacent to a .For each S ⊆ V , N ( S ) = S a ∈ S N ( a ) and N [ S ] = N ( S ) S S . A setof vertices S in G is a dominating set , if N [ S ] = V . The dominatingnumber , γ ( G ), of G is the minimum cardinality of a dominating set of G ([6]). Note that a graph whose vertices-set is empty is a null graph and a graph whose edge-set is empty is an empty graph.A module M is said to be a comultiplication R -module if for everysubmodule N of M there exists an ideal I of R such that N = Ann M ( I ).Also an R -module M is comultiplication module if and only if for eachsubmodule N of M , we have N = (0 : M Ann R ( N )) ([2]).In this article, we introduce and study the sum large graph ´ G ( M ) of M , where M is a comultiplication module. In section 2, we give thedefinition of ´ G ( M ) and consider some basic results on the structureof this graph. In Theorems 2.6 and 2.7, we provide some useful char-acterization about ´ G ( M ). In Theorem 2.9, it is shown that if ´ G ( M )is connected, then diam ( ´ G ( M ))
2. Also we prove that if ´ G ( M )contains a cycle, then g ( ´ G ( M )) = 3 (Theorem 2.10). Moreover, itis proved that if ´ G ( M ) is a connected graph, then ´ G ( M ) has no cutvertex (Theorem 2.11). Finally, in section 3, we investigate the cliquenumber, dominating number, and independence number of this graph. HE LARGE SUM GRAPH 3 Basic properties of ´ G ( M ) Definition 2.1.
Let M be an R -module. We define the large sum graph ´ G ( M ) of M with all non-large non-zero submodules of M as verticesand two distinct vertices N, K are adjacent if and only if N + K is anon-large submodule of M .A non-zero submodule S of M is said to be second if for each a ∈ R ,the endomorphism of M given by multiplication by a is either surjec-tive or zero. This implies that Ann R ( N ) is a prime ideal of R ([7]).The next lemma plays a key role in the sequel. Lemma 2.2.
Let M be a non-zero comultiplication R -module. (a) Every non-zero submodule of M contains a minimal submoduleof M . In particular, M in ( M ) = ∅ . (b) Let N be a submodule of M . Then N is a large submodule of M if and only if Soc ( M ) ⊆ N . (c) Let
N, K be submodules of M and let S be a second submoduleof M with S ⊇ N + K . Then S ⊇ N or S ⊇ K .Proof. (a) See [3, Theorem 3.2].(b) Let N be a large submodule of M . Assume to the contrary that Soc ( M ) * N . Then for each S j ∈ M in ( M ), we have S j * N . Since N is a large submodule of M , N ∩ S j = 0. Since S j is a minimalsubmodule of M and N ∩ S j ⊆ S j , we have S j ∩ N = S j , which isa contradiction. Conversely, suppose to the contrary that N is not alarge submodule of M . Then there exists a submodule K of M suchthat N ∩ K = 0. By part (a), there exists a minimal submodule L of M such that L ⊆ K . So we have L ∩ N ⊆ K ∩ N = 0, which impliesthat L * N , a contradiction. Thus N is a large submodule of M .(c) See [4, Theorem 2.6]. (cid:3) In the rest of this paper, we assume that M is a non-zero comulti-plication R -module. We recall that M in ( M ) = ∅ by Lemma 2.2 part(a). Lemma 2.3.
Let M be an R -module with M in ( M ) = { S i } i ∈ I , where | I | > , and let Λ be a non-empty proper finite subset of I . Then P λ ∈ Λ S λ is non-large submodule of M . H. ANSARI-TOROGHY AND F. MAHBOOBI-ABKENAR
Proof.
Let P λ ∈ Λ S λ be a large submodule of M and let j ∈ I \ Λ. Thenby Lemma 2.2 (b), S j ⊆ P λ ∈ Λ S λ . Since S j is a second submodule of M , by Lemma 2.2 (c), S j ⊆ S λ for some λ ∈ Λ, a contradiction. (cid:3)
We recall that an R -module M is said to be finitely cogenerated if for every set { M λ } λ ∈ Λ of submodules of M , ∩ λ ∈ Λ M λ = 0 implies ∩ ni =1 M λ i = 0 for some positive integer n ([1]). Moreover, an R -module M is said to be cocyclic if Soc(M) is a large and a simple submoduleof M ([7]). Proposition 2.4.
Let M be an R -module. Then ´ G ( M ) is a null graphif and only if M is a cocyclic module.Proof. This is straightforward. (cid:3)
Note that all definitions graph theory are for non-null graphs [5]. Soin the rest of this paper we assume that ´ G ( M ) is a non-null graph. Lemma 2.5.
Let M be an R -module. Then M is uniform if and onlyif M is a cocyclic R -module.Proof. This is obvious. (cid:3)
Theorem 2.6.
Let M be an R -module. Then ´ G ( M ) is an empty graphif and only if M in ( M ) = { S , S } such that MS and MS are finitelycogenerated uniform R -modules.Proof. Let ´ G ( M ) be an empty graph. If | M in ( M ) | >
2, then byLemma 2.3, S , S ∈ M in ( M ) are adjacent, a contradiction. Thus M in ( M ) = { S , S } . Now we claim that S + S S is a minimal submod-ule of MS because S S ∩ S ≃ S + S S , where S ∩ S = 0. We show that S + S S is the only minimal submodule of MS . Suppose that KS is a minimal sub-module of MS . If KS is a large submodule of MS , then S + S S = Soc ( M ) S ⊆ KS which implies that S + S S = KS , a contradiction. Now assume that KS is a non-large submodule of MS . Then K is a non-large submodule of M . So we have K + S = K M (i.e., K is a non-large submod-ule of M ) which follows that K and S are adjacent, a contradiction.Thus S + S S is the only minimal submodule of MS , and therefore MS isa uniform module. Since Soc ( MS ) = S + S S is a simple module, MS isa finitely cogenerated module by [1, Proposition 10.7]. Conversely, let M in ( M ) = { S , S } . Then clearly, S and S are not adjacent. We HE LARGE SUM GRAPH 5 claim that there is no vertex N = S , S . Assume to the contrarythat N is a vertex of ´ G ( M ). By Lemma 2.2 (a), S ⊆ N or S ⊆ N .Without loss of generality we can assume that S ⊆ N . One can seethat S + S S is a minimal submodule of MS . Since MS is a cocyclic moduleby lemma 2.5, for any submodule NS , we have S + S S = Soc ( MS ) ⊆ NS .Thus each submodule N of M is a large submodule by Lemma 2.2 (b),a contradiction. Hence ´ G ( M ) is an empty graph. (cid:3) Theorem 2.7.
Let M be an R -module. The following statements areequivalent. (i) ´ G ( M ) is not connected. (ii) | M in ( M ) | = 2(iii) ´ G ( M ) = ´ G ∪ ´ G , where ´ G and ´ G are complete and disjointsubgraphs.Proof. ( i ) ⇒ ( ii ) Assume to the contrary that | M in ( M ) | >
2. Since´ G ( M ) is not connected, we can consider two components ´ G , ´ G and N, K two submodules of M such that N ∈ ´ G and K ∈ ´ G . Choose S , S ∈ M in ( M ) such that S ⊆ N and S ⊆ K . If S = S , then N − S − K is a path, a contradiction. So we can assume that S = S .Since M in ( M ) > S + S is a non-large submodule of M by Lemma2.3. Thus N − S − S − K is a path between ´ G and ´ G , a contradiction.Therefore, | M in ( M ) | = 2.( ii ) ⇒ ( iii ) Let M in ( M ) = { S , S } . Set ´ G j := { N ≤ M | N ⊆ S j and N E M } , where j = 1 ,
2. Assume that
N, K ∈ ´ G . We claimthat N and K are adjacent. Otherwise, If N + K E M , then S + S = Soc ( M ) ⊆ N + K by Lemma 2.2 (b). Now by using Lemma 2.2 (c), S + S ⊆ N or S + S ⊆ K . So N or K are large submodules of M ,a contradiction. By using similar arguments for ´ G , we can concludethat ´ G , ´ G are complete subgraphs of ´ G ( M ). We claim that thesetwo subgraphs are disjoint. Assume to the contrary that N ∈ ´ G and N ∈ ´ G are adjacent. Then Soc ( M ) = S + S ⊆ N + N whichimplies that N + N is a large submodule of M by Lemma 2.2 (b), acontradiction.( iii ) ⇒ ( i ) This is obvious. (cid:3) Remark . The condition that “ M is a comultiplication module” cannot be omitted in Theorem 2.7. For example, let M = Z ⊕ Z be a Z -module and let N := (0 , Z , N := (0 , Z , N := (1 , Z , N :=(1 , Z , and N := (1 , Z . Then V ( ´ G ( M )) = { N , N , N , N , N } H. ANSARI-TOROGHY AND F. MAHBOOBI-ABKENAR and
M in ( M ) = { N , N , N } . Thus | M in ( M ) | > G ( M ) is not aconnected graph. Theorem 2.9.
Let ´ G ( M ) be a connected graph. Then diam ( ´ G ( M )) .Proof. Let N and K be two vertices of ´ G ( M ) such that they are notadjacent. By Lemma 2.2 (a), there exist two minimal submodules S , S of M such that S ⊆ N and S ⊆ K . If N + S M , then N − S − K isa path. So d ( N, K ) = 2. Similarly, if K + S M , then d ( N, K ) = 2.Now assume that N + S E M and K + S E M . By Theorem2.7, | M in ( M ) | ≥ G ( M ) is a connected graph. Let S be aminimal submodule of M such that S = S , S . Thus by Lemma 2.2(b), we have S ⊆ Soc ( M ) ⊆ N + S and S ⊆ Soc ( M ) ⊆ K + S .Now Lemma 2.2 (c) shows that S ⊆ N and S ⊆ K . Hence we have N − S − K . Therefore, d ( N, K ) = 2. (cid:3)
Theorem 2.10.
Let M be an R -module and ´ G ( M ) contains a cycle.Then g ( ´ G ( M )) = 3 .Proof. If | M in ( M ) | = 2, then ´ G ( M ) = ´ G ∪ ´ G , where ´ G and ´ G arecomplete disjoint subgraphs by Theorem 2.7. Since ´ G ( M ) contains acycle and ´ G , ´ G are disjoint complete subgraphs, g ( ´ G ( M )) = 3. Nowassume that | M in ( M ) | ≥ S , S , and S ∈ M in ( M ). ByLemma 2.3, S − S − S − S is a cycle. Hence g ( ´ G ( M )) = 3. (cid:3) A vertex a in a connected graph G is a cut vertex if G − { a } isdisconnected. Theorem 2.11.
Let M be an R -module. If ´ G ( M ) is a connected graph,then ´ G ( M ) has no cut vertex.Proof. Assume on the contrary that there exists a vertex N ∈ V ( ´ G ( M ))such that ´ G ( M ) \ N is not connected. Thus there exist at leat twovertices K, L such that N lies in every path between them. By Theorem2.9, the shortest path between K and L is length of two. So we have K − N − L . Firstly, we claim that N is a minimal submodule of M .Otherwise, there exists a minimal submodule S of M such that S ⊂ N by Lemma 2.2 (a). Since S + K ⊆ N + K and N + K E M , we have S + K M . By similar arguments, S + L is a non-large submodule of M . Hence K − S − L is a path in ´ G ( M ) \ N , a contradiction. Thus N is a minimal submodule of M . Now we claim that there is a minimalsubmodule S i = N such that S i * K . Suppose on the contrary that S i ⊆ K for each S i ∈ M in ( M ). So we have Soc ( M ) ⊆ K + N . This HE LARGE SUM GRAPH 7 implies that K + N is a large submodule of M by Lemma 2.2 (b),a contradiction. Similarly, there exits a minimal submodule S j = N of M such that S j * L . Note that for each S t ∈ M in ( M ), we have S t ⊆ K + L because K + L is a large submodule of M . So S t ⊆ K or S t ⊆ L by Lemma 2.2 (c). Now let N = S i , S j ∈ M in ( M ) such that S i * K and S j * L (Note that S i = S j ). Hence we have S i ⊆ L and S j ⊆ K . This implies that K − S i − S j − L is a path in ´ G ( M ) \ L , acontradiction. (cid:3) Theorem 2.12.
Let M be an R -module. Then ´ G ( M ) can not be acomplete n -partite graph.Proof. Suppose on the contrary that ´ G ( M ) is a complete n -partitegraph with parts U , U , ..., U n . By Lemma 2.3, for every S i , S j ∈ M in ( M ), S i and S j are adjacent. Hence each U i contains at mostone minimal submodule. By Pigeon hole principal, | M in ( M ) | n .Now we claim that | M in ( M ) | = t where t < n . Let S i ∈ V i , foreach i (1 ≤ i ≤ t ). Then V t +1 contains no minimal submodule of M . By Lemma 2.3, Σ j = i S j is a non-large submodule of M . Clearly,Σ j = i S j and S i are not adjacent because Soc ( M ) = Σ j = i S j + S i . HenceΣ j = i S j ∈ V i . Let N be a vertex in V t +1 . Then by Lemma 2.2 (a),there exists S k ∈ M in ( M ) such that S k ⊆ N . So N and S k are ad-jacent, where S k ∈ V k . Since ´ G ( M ) is a complete n -partite graph, N adjacent to all vertices in V k . So N and Σ j = k S j are adjacent. How-ever, Soc ( M ) = S k + Σ j = k S j which implies that N + Σ j = k S j E M by Lemma 2.3, a contradiction. Hence | M in ( M ) | = t . Now set K := Σ i =3 S i . By Lemma 2.3, K is a non-large submodule of M .Since K + S = Σ i =2 S i M , K and S are adjacent. Similarly, K isadjacent to S . Thus K V , V . Furthermore, K + S i = K M foreach i (1 ≤ i ≤ n ). Hence K is adjacent to all minimal submodules S i of M . So for each i (1 ≤ i ≤ n ), K V i , a contradiction. (cid:3) Proposition 2.13.
Let M be an R -module with | M in ( M ) | < ∞ . Thenwe have the following. (i) There is no vertex in ´ G ( M ) which is adjacent to every othervertex. (ii) ´ G ( M ) can not be a complete graph.Proof. (i) Assume on the contrary that there exists a submodule N ∈ V ( ´ G ( M )) such that N is adjacent to all vertices of ´ G ( M ). By Lemma2.2 (a), there is a minimal submodule S i ∈ M in ( M ) such that S i ⊆ N .Now set K := Σ j = i S j . Clearly, K M by Lemma 2.3. Since N isadjacent to all other vertices of ´ G ( M ), N + K is a non-large submodule H. ANSARI-TOROGHY AND F. MAHBOOBI-ABKENAR of M . However, Soc ( M ) = Σ j = i S j + S i ⊆ N + K which shows that N + K E M by Lemma 2.2 (b), a contradiction.(ii) This follows from (i). (cid:3) A vertex of a graph G is said to be pendent if its neighbourhoodcontains exactly one vertex. Theorem 2.14.
Let M be an R -module. Then we have the following. (i) ´ G ( M ) contains a pendent vertex if and only if | M in ( M ) | = 2 and ´ G ( M ) = ´ G ∪ ´ G , where ´ G , ´ G are two disjoint completesubgraphs and | V ( ´ G i ) | = 2 for some i = 1 , . (ii) ´ G ( M ) is not a star graph.Proof. (i) Let N be a pendent vertex of ´ G ( M ). Assume on the contrarythat | M in ( M ) | ≥
3. Clearly, for each S i ∈ M in ( M ), S i is adjacent toevery other minimal submodules of M . So deg ( S i ) ≥
2. Thus N is nota minimal submodule of M . By Lemma 2.2 (a), there exists a minimalsubmodule of S of M such that S ⊆ N . Note that the only vertexwhich is adjacent to N is S because deg ( N ) = 1. Hence there is nominimal submodule S i = S such that S i ⊆ N . Moreover, N + S is alarge submodule of M . So by Lemma 2.2 (b), S j ⊆ Soc ( M ) ⊆ N + S ,for each S j = S , S . This implies that S j ⊆ N by Lemma 2.2 (c),a contradiction. Hence | M in ( M ) | = 2. By Theorem 2.7, ´ G ( M ) =´ G ∪ ´ G , where ´ G and ´ G are disjoint complete subgraphs. This iseasy to see that | V ( ´ G i ) | = 2. The converse is straightforward.(ii) Suppose that ´ G ( M ) is a star graph. Then ´ G ( M ) has a pendentvertex. So by part (i), we have | M in ( M ) | = 2. Thus ´ G ( M ) isnot a connected graph by Theorem 2.7, a contradiction. (cid:3) Theorem 2.15.
Let M be an R -module. (i) Let N , K be two vertex of ´ G ( M ) such that N ⊆ K . Then deg ( N ) ≤ deg ( K ) . (ii) Let ´ G ( M ) be a r -regular graph. Then | M in ( M ) | = 2 and | V ( ´ G ( M )) | = 2 r + 2 .Proof. (i) Let N, K ∈ V ( ´ G ( M )) such that N ⊆ K . Let L be a vertexof ´ G ( M ) such that L is adjacent to N . Thus N + L is a non-largesubmodule of M and so that K + L is a non-large submodule of M .So L is adjacent to K . Therefore, deg ( N ) ≤ deg ( K ). HE LARGE SUM GRAPH 9 (ii) Suppose on the contrary that | M in ( M ) | ≥
3. By using Lemma2.3 and our assumption,
M in ( M ) is a finite set. Next for S , S ∈ M in ( M ), we have deg ( S ) ≤ deg ( S + S ) by part(a). We claim that deg ( S + S ) = deg ( S ). Otherwise, if N := Σ j =2 S j , then N is adjacent to S . However, N is not ad-jacent to S + S , a contradiction. So r < deg ( S + S ), whichis a contradiction. Thus | M in ( M ) | ≤
2. If | M in ( M ) | = 1,then ´ G ( M ) is null graph, a contradiction. Thus | M in ( M ) | = 2and so that by Theorem 2.6 ´ G ( M ) = ´ G ∪ ´ G , where ´ G , ´ G are disjoint complete subgraphs. Set M in ( M ) = { S , S } and S i ∈ G i . Since ´ G ( M ) is a r -regular graph, | V ( ´ G i ) | = r + 1 for i = 1 ,
2. Hence we have V ( ´ G ( M )) = 2 r + 2. (cid:3) clique number, dominating number, and independencenumber In this section, we obtain some results on the clique, dominating,and independence numbers of ´ G ( M ). Proposition 3.1.
Let M be an R -module. Then the following hold. (i) Let ´ G ( M ) be a non-empty graph. Then ω ( ´ G ( M )) ≥ | M in ( M ) | . (ii) Let ´ G ( M ) be an empty graph. Then ω ( ´ G ( M )) = 1 if and onlyif M in ( M ) = { S , S } , where S and S are finitely cogenerateduniform R -modules. (iii) If ω ( ´ G ( M )) < ∞ , then ω ( ´ G ( M )) ≥ | Min ( M ) |− − .Proof. (i) If | M in ( M ) | = 2, then ω ( ´ G ( M )) ≥ | M ax ( M ) | ≥
3. Then by Lemma 2.3, the subgraph of ´ G ( M ) withthe vertex set of { S i } S i ∈ Min ( M ) is a complete subgraph of ´ G ( M ). So ω ( ´ G ( M )) ≥ | M in ( M ) | .(ii) This follows directly from Theorem 2.6.(iii) Since ω ( ´ G ( M )) < ∞ , we have | M in ( M ) | < ∞ by part (i), (ii).Let M in ( M ) = { S , ..., S t } . For each 1 ≤ i ≤ t , consider A i = { S , ..., S i − , S i +1 , S t } .Now let P ( A i ) be the power set of A i and for each X ∈ P ( A i ),set S X = T S j ∈ X S j for 1 ≤ j ≤ t . The subgraph of ´ G ( M ) withthe vertex set { S X } X ∈ P ( A i ) \{∅} is a complete subgraph of ´ G ( M )by Lemma 2.3. It is clear that |{ S X } X ∈ P ( A i ) \{∅} | = 2 | Min ( M ) |− −
1. Thus ω ( ´ G ( M )) ≥ | Min ( M ) |− − (cid:3) Remark . Note that the condition “ M is a comultiplication module”is necessary in Proposition 3.1. For example, let M = Z ⊕ Z be as a Z -module which is not a comultiplication module. Then ω ( ´ G ( M )) = 2but | M in ( M ) | = 3. Theorem 3.3.
Let M be an R -module. Then γ ( ´ G ( M )) ≤ . In par-ticular, if | M in ( M ) | < ∞ , then γ ( ´ G ( M )) = 2 .Proof. Clearly, | M in ( M ) | ≥ G ( M ) is a non-null graph. Con-sider S = { S , S } , where S , S ∈ M in ( M ). Let N be a vertex of´ G ( M ). We claim that N is adjacent to S or S . If S ⊆ N or S ⊆ N ,then the claim is true. Now assume that S * N and S * N . Inthis case, we also claim that N is adjacent to S or S . Withoutloss of generality, we can assume that N is not adjacent to S . So S ⊆ Soc ( M ) ⊆ N by Lemma 2.2 (b). This shows that S ⊆ N ,which is a contradiction. By similar arguments, we can show that N is adjacent to S . Thus γ ( ´ G ( M )) ≤
2. The last assertion follows fromTheorem 2.13. (cid:3)
Theorem 3.4.
Let M be an R -module and | M in ( M ) | < ∞ . Then α ( ´ G ( M )) = | M in ( M ) | .Proof. Let
M in ( M ) = { S , ..., S n } . It is easy to see that { Σ nj =1 ,j = i S j } ni =1 is an independent set. So α ( ´ G ( M )) ≥ n . Now let α ( ´ G ( M )) = m and S = { N , ..., N m } be a maximal independent set. We claim that m = n .Assume on the contrary that m > n . Let S t ∈ M in ( M ). Then by Pi-geon hole principal, there exist N i , N j ∈ M in ( M ), where 1 ≤ i, j ≤ n ,such that S t * N i and S t * N j . Thus by using Lemma 2.2 (c), we have S t * N i + N j . Since N i , N j ∈ S and S is an independent set, we have S t ⊆ Soc ( M ) ⊆ N i + N j . Then by Lemma 2.2 (c), S t ⊆ N i or S t ⊆ N j ,which is a contradiction. Hence α ( ´ G ( M )) = n . (cid:3) Acknowledgments
We would like to thank Dr. F. Farshadifar for some helpful comments.
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E-mail address : [email protected] Department of Mathematics,University of Guilan, P.O.41335-19141,Rasht, Iran.
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