Commuting graphs of inverse semigroups and completely regular semigroups

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.