Let M be a unitary left R-module where R is a ring. The inclu- sion submodule graph of a module M , denoted by I n(M ), is an undirected simple graph whose vertex set V (I n(M )), is a set of all non-trivial 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 I n(M ) and some algebraic properties of modules. In particular, we consider several properties of the graph I n(M ) such as con- nectivity, diameter and the 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 I n(M ).
