Commuting Involution Graphs in Classical Affne Weyl Groups

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