Characterizing the largest commutative (full and partial) transformation semigroups of certain types

Let $X$ be a finite set. Let $\mathcal{T}(X)$ be the transformation semigroup on $X$ and let $\mathcal{P}(X)$ be the partial transformation semigroup on $X$. This paper is a contribution to the problem of characterizing the largest commutative subsemigroups of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$). In the process of looking for these semigroups, we also characterize the largest commutative subsemigroups of idempotents of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$); as well as the largest commutative subsemigroups of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$) that contain a unique idempotent. We also provide an alternative way to determine the largest commutative nilpotent subsemigroups of $\mathcal{T}(X)$ (which were previously characterized by Cain, Malheiro and the present author); and we describe the largest commutative nilpotent subsemigroups of $\mathcal{P}(X)$. These results allow us to make conclusions regarding the clique numbers of the commuting graphs of $\mathcal{T}(X)$ and of $\mathcal{P}(X)$. We also determine their girths and knit degrees.