On global defensive k-alliances in zero-divisor graph of finite commutative rings

TL;DR

This paper investigates the properties of global defensive k-alliances within zero-divisor graphs of finite commutative rings, extending and generalizing previous work for specific cases.

Contribution

It generalizes existing results on global defensive alliances in zero-divisor graphs to broader cases, improving upon prior research by Muthana and Mamouni.

Findings

Established new bounds and properties for global defensive k-alliances.

Provided examples illustrating the scope and limitations of the results.

Generalized previous specific case results to broader classes of rings.

Abstract

The global defensive $k$-alliance is a very well studied notion in graph theory, it provides a method of classification of graphs based on relations between members of a particular set of vertices. In this paper we explore this notion in zero-divisor graph of commutative rings. The established results generalize and improve recent work by Muthana and Mamouni who treated a particular case for $k=-1$ known by the global defensive alliance. Various examples are also provided which illustrate and delimit the scope of the established results.