Commuting graphs of completely simple semigroups

TL;DR

This paper explores the structure and properties of commuting graphs of completely simple semigroups, providing new characterizations and bounds for their graph-theoretic parameters.

Contribution

It introduces a detailed analysis of commuting graphs of Rees matrix semigroups over groups and characterizes which graphs can arise as commuting graphs of completely simple semigroups.

Findings

Determined bounds for diameter, clique number, girth, chromatic number, and knit degree.

Characterized the graphs that can be commuting graphs of completely simple semigroups.

Restricted possible values for properties of commuting graphs of groups.

Abstract

We describe the commuting graph of a Rees matrix semigroup over a group and investigate its properties: diameter, clique number, girth, chromatic number and knit degree. The maximum size of a commutative subsemigroup of a Rees matrix semigroup over a group is presented, and its largest commutative subsemigroups are exhibited. We use the knowledge we obtained from the commuting graph of this semigroup construction to deduce results regarding the properties of commuting graphs of completely simple semigroups. We also characterize the graphs that arise as commuting graphs of completely simple semigroups. In the process of obtaining these results we are also able to restrict the possible values for some properties of commuting graphs of groups.