The Zariski topology-graph on the maximal spectrum of modules over commutative rings

Let M be a module over a commutative ring and let Spec(M) (resp. Max(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 Max(M) as the induced subspace topology. For any non-empty subset T of Max(M), we introduce a new graph G(\tau^{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. Max(M)) by using the graph theoretical tools.