Commuting graphs of inverse semigroups and completely regular semigroups

TL;DR

This paper investigates the properties of commuting graphs in specific classes of semigroups, exploring whether certain graph properties can be realized within these classes.

Contribution

It provides a systematic study of the possible values of girth, clique number, chromatic number, and knit degree for commuting graphs of Clifford, inverse, and completely regular semigroups.

Findings

Characterization of commuting graph properties for different semigroup classes

Existence results for semigroups with prescribed commuting graph properties

Insights into the relationship between semigroup structure and graph invariants

Abstract

The general ideal of this paper is to answer the following question: given a numerical property of commuting graphs, a class of semigroups $\mathcal{C}$ and $n\in\mathbb{N}$, is it possible to find a semigroup in $\mathcal{C}$ such that the chosen property is equal to $n$? We study this question for the classes of Clifford semigroups, inverse semigroups and completely regular semigroups. Moreover, the properties of commuting graphs we consider are the girth, clique number, chromatic number and knit degree.