Commuting involution graphs for sporadic simple groups

Let K <= G <= Aut(K), where K is one of the 26 sporadic finite simple groups, and let t is an element of G be an involution and X = t(G). The commuting involution graph C(G, X) has X as its vertex set with two distinct elements of X joined by an edge whenever they commute in G. For most of the sporadic simple groups, we compute the diameter of C(G, X) and give detailed information about the elements at a given distance from a fixed involution t. (c) 2007 Elsevier Inc. All rights reserved.