Total graph of a module with respect to singular submodule

Abstract

Let R be a commutative ring with unity andM be an R-module. We introduce the total graph of a module M with respect to singular submodule Z(M) of M as an undirected graph T(Γ(M)) with vertex set as M and any two distinct vertices x and y are adjacent if and only if x + y ∈ Z(M). We investigate some properties of the total graph T(Γ(M)) and its induced subgraphs Z(Γ(M)) and Z(Γ(M)). In some aspects, we have noticed some sort of finiteness.