On the small intersection graph of submodules of a module

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 Γ(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 Γ(M) to the algebraic properties of the module M. We determine the diameter and the girth of Γ(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 Γ(M) is complemented.