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The Weakly Nilpotent Graph of a Commutative Ring

  Published:2017-01-26
 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
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Abstract

Let $R$ be a commutative ring with non-zero 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 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 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 $R$.
Keywords: weakly nilpotent graph, zero-divisor graph, diameter, girth weakly nilpotent graph, zero-divisor graph, diameter, girth
MSC Classifications: 05C15, 16N40, 16P20 show english descriptions Coloring of graphs and hypergraphs
Nil and nilpotent radicals, sets, ideals, rings
Artinian rings and modules
05C15 - Coloring of graphs and hypergraphs
16N40 - Nil and nilpotent radicals, sets, ideals, rings
16P20 - Artinian rings and modules
 

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