The zero-divisor graphs of semirings

TL;DR

This paper investigates the properties of zero-divisor graphs in rings and semirings, establishing connectivity, diameter bounds, and characterizations of specific graph structures, including acyclic, cyclic, and complete multipartite graphs.

Contribution

It provides a comprehensive characterization of zero-divisor graphs in semirings, including conditions for acyclicity, cyclicity, girth, and specific graph classes, extending previous ring theory results.

Findings

All zero-divisor graphs of semirings are connected with diameter ≤ 3.

Characterization of all acyclic zero-divisor graphs of semirings.

Identification of conditions for cyclic zero-divisor graphs and their girths.

Abstract

In this paper we study zero--divisor graphs of rings and semirings. We show that all zero--divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than or equal to 4. We find all possible cyclic zero-divisor graphs over commutative semirings having at most one 3-cycle, and characterize all complete $k$-partite and regular zero-divisor graphs. Moreover, we characterize all additively cancellative commutative semirings and all commutative rings such that their zero--divisor graph has exactly one 3-cycle.