Domination and Total Domination Numbers in Zero-divisor Graphs of Commutative Rings

TL;DR

This paper investigates the domination and total domination numbers of zero-divisor graphs of commutative rings, revealing that these numbers are generally equal except for a specific case involving the ring rac{rac{Z}_2 imes D.

Contribution

It determines the domination and total domination numbers for zero-divisor graphs of commutative rings, highlighting a unique exception involving rac{rac{Z}_2 imes D.

Findings

Domination and total domination numbers are equal for all zero-divisor graphs of commutative rings except for rac{rac{Z}_2 imes D.

In the exceptional case, rac{rac{Z}_2 imes D}, the domination number is 1 and the total domination number is 2.

The paper characterizes the domination properties of zero-divisor graphs in relation to the structure of the underlying rings.

Abstract

Zero-divisor graphs of commutative rings are well-represented in the literature. In this paper, we consider dominating sets, total dominating sets, domination numbers and total domination numbers of zero-divisor graphs. We determine the domination and total domination numbers of zero-divisor graphs are equal for all zero-divisor graphs of commutative rings except for $\mathbb{Z}_2 \times D$ in which $D$ is a domain. In this case, $\gamma(\Gamma(\mathbb{Z}_2 \times D)) = 1$ and $\gamma_t(\Gamma(\mathbb{Z}_2 \times D)) = 2$.