Maximal submodule graph of a module

Abstract

Let U be a left R-module where R is a commutative ring with identity. The maximal submodule graph MG(U) of proper R-submodules of U is an undirected graph defined as follows: the vertex set is the set of all proper R-submodules of U, and there is an edge between two distinct vertices N and L if and only if N + L is a maximal submodule of U. We study these graphs to relate the combinatorial properties of MG(U) to the algebraic properties of the R-module U. We study connectedness for MG(U). We investigated some properties of MG(U) such as, diameter, girth, and clique number.