Let R be a ring with identity and M be a unitary left Rmodule. The inclusion submodule graph of a module M, denoted by In(M), is an undirected simple graph whose vertex set V (In(M)) is a set of all nontrivial submodules of M and there is an edge between two distinct vertices X and Y if and only if X ⊂ Y or Y ⊂ X. In this paper, we investigate connections between the graph-theoretic properties of In(M) and some algebraic properties of modules. In particular, we consider several properties of the graph In(M), such as connectivity, diameter and girth. Also we obtain some independent sets and universal vertices of this graph. We characterize some modules for which the inclusion submodule graphs are connected, complete and null. Finally, we study the clique number and the chromatic number of In(M).
