On perfectness of dot product graph of a commutative ring

Abstract

Let $A$ be a commutative ring with nonzero identity, and $1leq n<infty$ be an integer, and $R=Atimes Atimescdotstimes A$ ($n$ times). The total dot product graph of $R$ is the (undirected) graph $TD(R)$ with vertices $R^*=Rsetminus {(0,0,dots,0)}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xcdot y=0in A$ (where $xcdot y$ 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 $ZD(R)$.