Document Type : Research Paper

**Authors**

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

**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 $ZD(R)$.

$R=A\times A\times\cdots\times A$ ($n$ 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**

[2] D. F. Anderson, P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434-447.

[3] D. F. Anderson, On the diameter and girth of a zero-divisor graph, II. Houston J. Math, (2008) 34:361-371.

[4] A. Badawi, On the Annihilator Graph of a commutative ring, Communications in Algebra, 42 (2014) 108-121.

[5] A. Badawi, On the dot product graph of a commutative ring, Communications in Algebra, 43 (2015) 43-50.

[6] R. Diestel, Graph Theory NY, USA: Springer-Verlag, (2000).

[7] D.B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, Upper Saddle River (2001).

November 2019

Pages 1-7