On the global powerful alliance number of zero-divisor graphs of finite commutative rings

Abstract

Let Gamma = (V, E) be a finite undirected graph without loops or multiple edges. A non-empty set of vertices S subset of V is called powerful alliance if for every vertex u is an element of N[S], |N[u] boolean AND S| >= |N[u] (S) over bar|. A powerful alliance dominating set is called global. The global powerful alliance number gamma(ap)(Gamma) is defined as the minimum cardinality among all global powerful alliances. In this paper, we initiate the study of the global powerful alliance number of zero-divisor graphs Gamma(R) with R is a finite commutative ring. Hence, we calculate gamma(ap)(Gamma(R)) for some usual kind of finite rings. As application, we give the global powerful alliance number of all zero-divisor graphs of finite commutative rings of order <= 7.