The quest for the clique numbers of commuting graphs of semigroups

The commuting graph of a finite non-commutative semigroup $S$ is the simple graph whose vertices are the non-central elements of $S$ and where two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$.

In this talk we will go on a quest to find the clique numbers of the commuting graphs of (full and partial) transformation semigroups. This epic journey will start with commuting graphs, it will then move on to maximum-order commutative subsemigroups of certain types, with a quick visit to trees of commutative transformation semigroups, and then back to commuting graphs.