Commuting graphs and semigroup constructions
Tânia Paulista
TL;DR
This paper investigates how commuting graphs associated with semigroups behave under two standard constructions: the zero-union and the direct product. It develops structural descriptions and exact formulae for key graph invariants, linking $\mathcal{G}(S)$ and $\mathcal{G}^*(S)$ of the constructed semigroup to those of the original factors. For zero-unions, the results show a dichotomy: if there is a single noncommutative component, $\mathcal{G}(S)$ mirrors that component; if there are multiple, $\mathcal{G}(S)$ becomes the graph join of the noncommutative parts, yielding diameter $2$ and additive behavior for clique and chromatic numbers, with detailed girth and knit-degree consequences. For direct products, the extended graph decomposes as a (direct/strong) product of the factors' extended graphs, and $\omega(\mathcal{G}(S))$, $\chi(\mathcal{G}(S))$, and knot-degree-type parameters are expressed in terms of the corresponding invariants of the factors. Overall, the work provides a systematic framework to compute and compare central graph invariants of commuting graphs under two fundamental semigroup constructions, enhancing our understanding of the algebra-graph correspondence.
Abstract
The aim of this paper is to see how commuting graphs interact with two semigroup constructions: the zero-union and the direct product. For both semigroup constructions, we investigate the diameter, clique number, girth, chromatic number and knit degree of their commuting graphs and, when possible, we exhibit the relationship between each one of these properties and the corresponding properties of the commuting graphs of the original semigroups.