Commuting graphs of completely 0-simple semigroups
Tânia Paulista
TL;DR
This work analyzes commuting graphs of completely $0$-simple semigroups by leveraging the $0$-Rees matrix representation $\\mathcal{M}_0[G; I,\\Lambda; P]$. It introduces the $0$-closure method to decide connectivity, identify components, and compute diameters of connected components, tying these to the connectivity of $\\mathcal{G}(I,\\Lambda, P)$. The authors derive explicit formulas and bounds for the clique number, girth, and chromatic number in terms of the pattern of zeros in $P$, and they characterize the knit degree via left paths. They further show that, for any $n \ge 2$, one can construct finite noncommutative completely $0$-simple semigroups with prescribed diameter, clique number, and chromatic number, thereby demonstrating a rich range of possible invariants. The paper concludes with open problems on exact chromatic numbers and realizable commuting graphs, inviting further structural exploration of these semigroups and their graphs.
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.