Let R be a ring with identity and M be a unitary left R- module. The co-intersection graph of proper submodules of M , Ω(M ) is an undirected simple graph whose vertices are non-trivial submodule of M in which two vertices N and K are joined by an edge, if and only if N + K = M . In this paper, we study several properties of Ω(M ). We prove that, if Ω(M ) is a path, then Ω(M ) ∼= P2 or Ω(M ) ∼= P3 . We show that, if Ω(M ) is a forest, then each component of Ω(M ) is complete or star graph. We determine the conditions under which Ω(M ) is weakly perfect. Moreover, we introduce the universal vertices and the dominating sets of Ω(M ) and their relationship with the non-trivial small submodules of M .
