Zero-divisor graphs of semirings with no S-vertices

Let R be a commutative semiring (ring) with identity ۱ \neq ۰‎. ‎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 \Gamma(R)‎, ‎using the notion of the S-vertices in connection with the nonexistence of S-vertices in \Gamma(R)‎. ‎We discuss when \Gamma(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 \Gamma(R).

کلیدواژه ها:

Complete bipartite graph‎ ، Weakly perfect graph‎ ، ‎r-partite graph‎ ، ‎Smarandache vertex (S-vertex) of a graph ، Smarandache zero-divisor