Let M be a module over a commutative ring R. The annihilatingsubmodule graph of M , denoted by AG(M ), is a simple graph in which a non-zero submodule N of M is a vertex if and only if there exists a non-zero proper submodule K of M such that N K = (0), where N K, the product of N and K, is denoted by (N : M )(K : M )M and two distinct vertices N and K are adjacent if and only if N K = (0). This graph is a submodule version of the annihilating-ideal graph and under some conditions, is isomorphic with an induced subgraph of the Zariski topology-graph G(τ T ) which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283-3296). In this paper, we study the domination number of AG(M ) and some connections between the graph-theoretic properties of AG(M ) and algebraic properties of module M . 2010 Mathematics Subject Classification. 13C13, 13C99, 05C75.

Let M be a module over a commutative ring R. In this paper, we continue our study about the quasi-Zariski topology-graph G(τ * T) which was introduced in (On the graph of modules over commutative rings, Rocky Mountain J. Math. 46(3) (2016), 1-19). For a non-empty subset T of Spec(M), we obtain useful characterizations for those modules M for which G(τ * T) is a bipartite graph. Also, we prove that if G(τ * T) is a tree, then G(τ * T) is a star graph. Moreover, we study coloring of quasi-Zariski topology-graphs and investigate the interplay between χ(G(τ * T)) and ω(G(τ * T)).

Let M be a module over a commutative ring R. The annihilatingsubmodule graph of M , denoted by AG(M ), is a simple undirected graph in which a non-zero submodule N of M is a vertex if and only if there exists a nonzero proper submodule K of M such that N K = (0), where N K, the product of N and K, is denoted by (N : M )(K : M )M and two distinct vertices N and K are adjacent if and only if N K = (0). This graph is a submodule version of the annihilating-ideal graph and under some conditions, is isomorphic with an induced subgraph of the Zariski topology-graph G(τ T ) which was introduced in [H. Ansari-Toroghy and S. Habibi, Comm. Algebra, 42(2014), 3283-3296]. In this paper, we study the domination number of AG(M ) and some connections between the graph-theoretic properties of AG(M ) and algebraic properties of module M .

Let M be a module over a commutative ring and let Spec(M ) (resp. M ax(M )) be the collection of all prime (resp. maximal) submodules of M . We topologize Spec(M ) with Zariski topology, which is analogous to that for Spec(R), and consider M ax(M ) as the induced subspace topology. For any non-empty subset T of M ax(M ), we introduce a new graph G(τ m T ), called the Zariski topology-graph on the maximal spectrum of M . This graph helps us to study the algebraic (resp. topological) properties of M (resp. M ax(M )) by using the graph theoretical tools.1 2 HABIBOLLAH ANSARI-TOROGHY AND SHOKOOFEH HABIBI a new graph G(τ T ), called the Zariski topology-graph. This graph helps us to study algebraic (resp.topological) properties of modules (resp. Spec(M)) by using the graphs theoretical tools.a non-empty subset of Spec(M ) and distinct vertices N and L are adjacent if and only if V (N ) ∪ V (L) = T . There exists a topology on M ax(M ) having Z m (M ) = {V m (N ) : N ≤ M } as the set of closed sets of M ax(M ), where V m (N ) = {Q ∈ M ax(M ) : (Q : M ) ⊇ (N : M )}. We denote this topology by τ m M . In fact τ m M is the same as the subspace topology induced by τ M on M ax(M ).In this paper, we define a new graph G(τ m T ), called the Zariski topology-graph on the maximal spectrum of M, where T is a non-empty subset of M ax(M ), and by using this graph, we study algebraic (resp. topological) properties of M (resp.a non-empty subset of M ax(M ) and distinct vertices N and L are adjacent if and only if V m (N ) V m (L) = T . Let T be a non-empty subset of M ax(M ). As τ m M is the subspace topology induced by τ M on M ax(M ), one may think that G(τ T ) and G(τ m T ) have the identical nature. But the Example 3.5 (case (1)) shows that this is not true and these graphs are different. Also the Example 3.5 (case (2)) shows that for a non-empty subset T ′ of Spec(M ), under a condition, G(τ T ′ ) can be regarded as a subgraph of G(τ m T ), where T = T ′ ∩M ax(M ). Besides, this case denotes that G(τ T ′ ) is not a subgraph of G(τ m T ), necessarily. Moreover, it is shown that G(τ m T ) can not appear as a subgraph of G(τ T ′ ), where T ⊆ T ′ ⊆ Spec(M ), in general (see Example 3.5 (case (3)). So, the results related to G(τ m T ), where T is a non-empty subset of M ax(M ), do not go parallel to those of G(τ T ′ ), where T ′ is a non-empty subset of Spec(M ), necessarily. Based on the above remarks, it is worth to study Max-graphs separately. For any pair of submodules N ⊆ L of M and any element m of M , we denote L/N and the residue class of m modulo N in M/N by L and m respectively. For a submodule N of M , the colon ideal of M into N is defined by (N : M ) = {r ∈ R : rM ⊆ N } = Ann(M/N ). Further if I is an ideal of R, the submodule (N : M I) is defined by {m ∈ M : Im ⊆ N }. Moreover, Z (resp. Q) denotes the ring of integers (resp. the field of rational numbers). The prime radical √ N is defined to be the intersection of all prime submodules of M containing N , and in case N is not contained in any prime submodule, √ N is def...

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