Commuting Involution Graphs in Classical Affine Weyl Groups

Abstract

In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let $W$ be a classical Weyl group of rank $n$, with $\tilde{W}$ its corresponding affine Weyl group. Our main result is that if $X$ is a conjugacy class of involutions in $\tilde{W}$, then the commuting involution graph $\mathcal{C}(\tilde{W},X)$ is either disconnected or has diameter at most $n+2$. This bound is known to hold for types ${\tilde{A}}_n$ and ${\tilde{C}}_n$, so the main work of this paper is to prove the theorem for types ${\tilde{B}}_n$ and ${\tilde{D}}_n$.