Co-intersection graph of submodules of a module

Let $M$ be a unitary left $R$-module where $R$ is a ring with identity. The co-intersection graph of proper submodules of $M$, denoted by $\Omega(M)$, is an undirected simple graph whose the vertex set $V(\Omega)$ is a set of all non-trivial submodules of $M$ and there is an edge between two distinct vertices $N$ and $K$ if and only if $N+K\neq M$. In this paper we investigate connections between the graph-theoretic properties of $\Omega(M)$ and some algebraic properties of modules . We characterize all of modules for which the co-intersection graph of submodules is connected. Also the diameter and the girth of $\Omega(M)$ are determined. We study the clique number and the chromatic number of $\Omega(M)$.