Commuting graphs of completely 0-simple semigroups

TL;DR

This paper investigates the properties of commuting graphs of completely 0-simple semigroups, providing methods to determine connectivity, diameter, clique number, girth, chromatic number, and knit degree, based on their structure as 0-Rees matrix semigroups.

Contribution

It introduces a comprehensive framework to analyze the commuting graphs of completely 0-simple semigroups, including connectivity criteria and bounds for various graph invariants.

Findings

Decided criteria for connectivity of commuting graphs.

Derived formulas for diameter and connected component diameters.

Established bounds for clique number, girth, and chromatic number.

Abstract

The aim of this paper is to study commuting graphs of completely $0$-simple semigroups, using the characterization of these semigroups as $0$-Rees matrix semigroups over a groups. We establish a method to decide whether the commuting graph of this semigroup construction is connected or not. If it is not connected, we also supply a way to identify the connected components of the commuting graph. We show how to obtain the diameter of the commuting graph (when it is connected) and the diameters of the connected components of the commuting graph (when it is not connected). Moreover, we obtain the clique number and girth of the commuting graph of such a semigroup, as well as two upper bounds (either of which can be the best in different situations) for its chromatic number. We also determine the knit degree of such a semigroup. Finally, we use the results regarding the properties of the commuting graph of a $0$-Rees matrix semigroup over a group to determine the set of possible values for the diameter, clique number, girth, chromatic number and knit degree of the commuting graph of a completely $0$-simple semigroup.