On global offensive alliance in zero-divisor graphs

Abstract

Let Γ(V, E) be a simple graph without loops nor multiple edges. A nonempty subset S ⊆ V is said a global offensive alliance if every vertex v ∈ V − S satisfies that δ S (v) ≥ δ S (v) + 1. The global offensive alliance number γ o (Γ) 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 Γ(R).