On the small intersection graph of submodules of a module

Abstract

Let $M$ be a unitary left $R$-module, where $R$ is a (not necessarily commutative) ring with identity. The small intersection graph of nontrivial submodules of $M$, denoted by $Gamma(M)$, is an undirected simple graph whose vertices are in one-to-one correspondence with all nontrivial submodules of $M$ and two distinct vertices are adjacent if and only if the intersection of corresponding submodules is a small submodule of $M$. In this paper, we investigate the fundamental properties of these graphs to relate the combinatorial properties of $Gamma(M)$ to the algebraic properties of the module $M$. We determine the diameter and the girth of $Gamma(M)$. We obtain some results for connectivity and planarity of these graphs. Moreover, we study orthogonal vertex, domination number and the conditions under which the graph $Gamma(M)$ is complemented.