The Weakly Nilpotent Graph of a Commutative Ring

Abstract

Let $R$ be a commutative ring with non-zero identity. In this paper, we introduce the weakly nilpotent graph of a commutative ring. The weakly nilpotent graph of $R$ denoted by ${{\Gamma }_{w}}(R)$ is a graph with the vertex set ${{R}^{\star }}$ and two vertices $x$ and $y$ are adjacent if and only if $x\,y\in N{{(R)}^{\star }}$, where ${{R}^{\star }}=R\backslash \{0\}$ and $N{{(R)}^{\star }}$ is the set of all non-zero nilpotent elements of $R$. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if ${{\Gamma }_{w}}(R)$ is a forest, then ${{\Gamma }_{w}}(R)$ is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of ${{\Gamma }_{w}}(R)$. Among other results, we show that for an Artinian ring $R$, ${{\Gamma }_{w}}(R)$ is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam $\overline{({{\Gamma }_{w}}(R))}$. Finally, we characterize all commutative rings $R$ for which $\overline{({{\Gamma }_{w}}(R))}$ is a cycle, where $\overline{({{\Gamma }_{w}}(R))}$ is the complement of the weakly nilpotent graph of $R$.