Abstract view
The Weakly Nilpotent Graph of a Commutative Ring


Published:20170126
Printed: Jun 2017
Sohiela Khojasteh,
Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran
Mohammad Javad Nikmehr,
Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
Abstract
Let $R$ be a commutative ring with nonzero identity. In this
paper, we introduced the weakly nilpotent graph of a commutative
ring. The weakly nilpotent graph of $R$ is denoted by $\Gamma_w(R)$
is a graph with the vertex set $R^{*}$ and two vertices $x$ and
$y$ are adjacent if and only if $xy\in N(R)^{*}$, where $R^{*}=R\setminus\{0\}$
and $N(R)^{*}$ is the set of all nonzero 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
ring, we determine $\operatorname{diam}(\overline{\Gamma_w(R)})$. 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 $R$.