Commuting Graphs of Boundedly Generated Semigroups

TL;DR

This paper explores the structure of commuting graphs of semigroups, showing that any star-free graph can be realized as such and analyzing properties like clique number and diameter within boundedly generated semigroups.

Contribution

It demonstrates that all star-free graphs are realizable as commuting graphs of semigroups and establishes bounds on diameters for certain classes of semigroups.

Findings

Any star-free graph is the commuting graph of some semigroup.

Constructed monomial semigroups with bounded generators can have arbitrary clique numbers.

The diameter of commuting graphs in certain semigroup classes is bounded by the number of generators plus two.

Abstract

Ara\'ujo, Kinyon and Konieczny (2011) pose several problems concerning the construction of arbitrary commuting graphs of semigroups. We observe that every star-free graph is the commuting graph of some semigroup. Consequently, we suggest modifications for some of the original problems, generalized to the context of families of semigroups with a bounded number of generators, and pose related problems. We construct monomial semigroups with a bounded number of generators, whose commuting graphs have an arbitrary clique number. In contrast to that, we show that the diameter of the commuting graphs of semigroups in a wider class (containing the class of nilpotent semigroups), is bounded by the minimal number of generators plus two. We also address a problem concerning knit degree.