Roman domination number of zero-divisor graphs over commutative rings

TL;DR

This paper investigates the Roman domination number of zero-divisor graphs over commutative rings, establishing bounds and exploring properties relevant to graph theory and algebraic structures.

Contribution

It introduces bounds for the Roman domination number of zero-divisor graphs and analyzes its properties in the context of commutative rings.

Findings

Bounds for the Roman domination number of zero-divisor graphs are established.

The paper explores the relationship between the structure of rings and the Roman domination number.

Properties of the Roman domination number in relation to graph transformations are analyzed.

Abstract

For a graph $G= (V, E)$, a Roman dominating function is a map $f : V \rightarrow \{0, 1, 2\}$ satisfies the property that if $f(v) = 0$, then $v$ must have adjacent to at least one vertex $u$ such that $f(u)= 2$. The weight of a Roman dominating function $f$ is the value $f(V)= \Sigma_{u \in V} f(u)$, and the minimum weight of a Roman dominating function on $G$ is called the Roman domination number of $G$, denoted by $\gamma_R(G)$. The main focus of this paper is to study the Roman domination number of zero-divisor graph $\Gamma(R)$ and find the bounds of the Roman domination number of $T(\Gamma(R))$.