Properties of co-intersection graph of submodules of a module

Let \(R\) be a ring with identity and \(M\) be a unitary left Rmodule. 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 \(N + K \neq M\). In this paper, we study several properties of \(Ω(M)\). We prove that, if \(Ω(M)\) is a path, then \(Ω(M) \cong P_2 \)or \(Ω(M) \cong P_3\). 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 \(M\).