On perfectness of dot product graph of a commutative ring

Document Type : Research Paper

Authors

Department of Mathematics, Islamic Azad University, Central Tehran Branch, P. O. Box 14168-94351, Iran

10.29252/as.2019.1399

Abstract

Let $A$ be a commutative ring with nonzero identity, and $1\leq n<\infty$ be an integer, and
$R=A\times A\times\cdots\times A$ ($n$ times). The total dot product graph of $R$ is the (undirected) graph $TD(R)$ with vertices $R^*=R\setminus \{(0,0,\dots,0)\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x\cdot y=0\in A$ (where $x\cdot y$ denote the normal dot product of $x$ and $y$).
 Let $Z(R)$ denote the set of all zero-divisors of $R$.  Then the zero-divisor dot product graph of $R$ is the induced subgraph $ZD(R)$ of $TD(R)$ with vertices  $Z(R)^*=Z(R)\setminus \{(0,0,\dots,0)\}$. It follows that if  $\Gamma(A)$ is not  perfect, then  $ZD(R)$ (and hence $TD(R)$) is not  perfect.
In this paper we investigate perfectness of the graphs $TD(R)$ and $ZD(R)$.

Keywords


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