On defensive alliance in zero-divisor graphs
Let [Formula: see text] be a finite undirected graph without loops or multiple edges. A non-empty set of vertices [Formula: see text] is called defensive alliance if every vertex in [Formula: see text] have at least one more neighbors inside of [Formula: see text] than it has outside of [Formula: see text]. A non-empty set of vertices [Formula: see text] is called a dominating set if every vertex not in [Formula: see text] is adjacent to at least one member of [Formula: see text]. A defensive alliance dominating set is called global. The global defensive alliance number [Formula: see text] is defined as the minimum cardinality among all global defensive alliances. In this paper, we initiate the study of the global defensive alliance number of zero-divisor graphs [Formula: see text] with [Formula: see text] is a finite commutative ring. Hence, we calculate [Formula: see text] for some usual kind of rings. We finish by a complete characterization of rings with [Formula: see text].