فهرست نشریات علمی دسترسی آزاد ایران

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 $ZD(R)$.