On perfectness of dot product graph of a commutative ring

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≤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).

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References

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Algebraic Structures and Their Applications
Volume 6, Issue 2
November 2019
Pages 1-7

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