INCLUSION SUBMODULE GRAPH OF A MODULE

Let M be a unitary left R-module where R is a ring with identity. 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 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 consider several properties of the graph In(M) such as connectivity, diameter and the girth. 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).