Partitioning zero-divisor graphs of finite commutative rings into global defensive alliances

Abstract

For a commutative ring R with identity, the zero-divisor graph of R, denoted \(\Gamma (R)\), is the simple graph whose vertices are the nonzero zero-divisors of R where two distinct vertices x and y are adjacent if and only if \(xy=0\). In this paper, we are interested in partitioning the vertex set of \(\Gamma (R)\) into global defensive alliances for a finite commutative ring R. This problem has been well investigated in graph theory. Here we connected it with the ring theoretical context. We characterize various commutative finite rings for which the zero-divisor graph is partitionable into global defensive alliances. We also give several examples to illustrate the scopes and limits of our results.