Abstract
Let Γ(V,E) be a simple connected graph with more than one vertex, without loops or multiple edges. A nonempty subset S⊆V is a global offensive alliance if every vertex v∈V−S satisfies that δS(v)≥δS¯(v)+1. The global offensive alliance numberγo(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 study the global offensive alliance number of the zero-divisor graph Γ(R).
Highlights
Alliances had never been explored in this type of graph until recently, in 2020, when Muthana and Mamouni initiated the study of the global offensive alliance number in the zero-divisor graph [26]
This paper is organized as follows: in Section 2.1, we give some results concerning the global offensive alliance number of the zero-divisor graph; for example, we give a characterization in terms of the global offensive alliance number for Γ( R) to be a complete graph
Our main goal is to calculate or provide sharp bounds to the global offensive alliance number of zero-divisor graphs for some kind of direct products of finite local rings with finite fields—in particular, to characterize when the global offensive alliance number over the ring formed by the direct product of Z2 with any ring R is | Z ( R)∗ | + 1
Summary
Alliances in graphs serve as a mathematical model for several practical and theoretical problems that have been appearing in the literature of different areas of knowledge, such as data structure [11], web communities [12], bioinformatics (study of the proteome and genome) [13], as well as defense systems [14]. Alliances had never been explored in this type of graph until recently, in 2020, when Muthana and Mamouni initiated the study of the global offensive alliance number in the zero-divisor graph [26]. This paper is organized as follows: in Section 2.1, we give some results concerning the global offensive alliance number of the zero-divisor graph; for example, we give a characterization in terms of the global offensive alliance number for Γ( R) to be a complete graph.
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