A generalization of total graphs of modules

Abstract

Let $R$ be a commutative ring, and let $M\neq 0$ be an $R$-module with a non-zero proper submodule $N$, where $N^{\star}=N-\{0\}$. Let $\Gamma_{N^{\star}}(M)$ denote the (undirected) simple graph with vertices $ \{x \in M -N\,|\,x+x^\prime \in N^{\star}$ for some $x\neq x' \in M-N \}$, where distinct vertices $x$ and $y$ are adjacent if and only if $x+y \in N^{\star}$. We determine some graph theoretic properties of $\Gamma_{N^{\star}}(M)$ and investigate the independence number and chromatic number.