On commuting graphs

The \emph{commuting graph} $\Gamma(S)$ of a semigroup (or a semiring) $S$ is the graph, whose vertex set is the set of all noncentral elements of $S$ and $x-y$ is an edge in $\Gamma(S)$ if $xy=yx$ and $x \ne y$. The commuting graphs are an illustrative way of describing centralizers of elements in certain algebraic structures. In the talk, we introduce commuting graphs and give some recent results on diameters of commuting graphs of different algebraic structures. The main part of the talk will be concentrating on commuting graphs of matrices over rings and semirings.