On global offensive alliance in zero-divisor graphs

TL;DR

This paper introduces the concept of global offensive alliances within zero-divisor graphs of finite commutative rings, establishing foundational properties and calculating the global offensive alliance number for these graphs.

Contribution

It initiates the study of global offensive alliance numbers in zero-divisor graphs of finite rings, a novel intersection of graph theory and ring theory.

Findings

Defined the global offensive alliance number for zero-divisor graphs.

Calculated the global offensive alliance number for specific classes of rings.

Established bounds and properties related to these alliances in zero-divisor graphs.

Abstract

Let $\G(V,E)$ be a simple graph without loops nor multiple edges. A nonempty subset $S \subseteq V$ is said a {\em global offensive alliance} if every vertex $v \in V- S$ satisfies that $\d_S(v) \geq \d_{\overline{S}}(v)+1$. The {\em global offensive alliance number} $\g^o(\Gamma)$ is defined as the minimum cardinality among all global offensive alliances. Let $R$ be a finite commutative ring with identity. In this paper, we initiate the study of the global offensive alliance number of the zero-divisor graph $\G(R)$.