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 Ω(M ), is an undirected simple graph whose the vertex set V (Ω) 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 = M . In this paper we investigate connections between the graph-theoretic properties of Ω(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 Ω(M ) are determined. We study the clique number and the chromatic number of Ω(M ).
