Zero-divisor graphs of semirings with no S-vertices

Abstract

Let R be a commutative semiring (ring) with identity 1 ≠ 0. A vertex a in a simple graph G is said to be a Smarandache vertex (or S-vertex for short) provided that there exist three distinct vertices x, y, and b (all different from a) in G such that x—a, a—b, and b—y are edges in G, but there is no edge between x and y. In this interdisciplinary subject, we investigate the interplay between the algebraic properties of the commutative semirings and their associated zero-divisor graphs, denoted by Γ(R), using the notion of the S-vertices in connection with the nonexistence of S-vertices in Γ(R). We discuss when Γ(R) is a complete bipartite graph together with some of its other graph-theoretic properties and their relation to the nonexistence of S-vertices of Γ(R).