Co-intersection graph of submodules of a module

Abstract

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 \(\Omega(M)\).