Commuting Involution Graphs in classical affine Weyl Groups

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