On perfectness of dot product graph of a commutative ring

Let A be a commutative ring with nonzero identity, and ۱\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 \{(۰,۰,\dots,۰)\}, and two distinct vertices x and y are adjacent if and only if x\cdot y=۰\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 \{(۰,۰,\dots,۰)\}. 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:

annihilator graph , Zero-divisor , Complete graph