On perfectness of dot product graph of a commutative ring

Let A be a commutative ring with nonzero identity, and 1≤n<∞ be an integer, and R=A×A×⋯×A (n times). The total dot product graph of R is the (undirected) graph TD(R) with vertices R∗=R∖{(0,0,…,0)}, and two distinct vertices x and y are adjacent if and only if x⋅y=0∈A (where x⋅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)∖{(0,0,…,0)}. It follows that if Γ(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).