Let K G Aut(K), where K is one of the 26 sporadic finite simple groups, and let t ∈ 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.

Let $G$ be a group and $S$ a non-empty subset of $G$. If $ab \notin S$ for any $a, b \in S$, then $S$ is called sum-free. We show that if $S$ is maximal by inclusion and no proper subset generates $\langle S\rangle$ then $|S|\leq 2$. We determine all groups with a maximal (by inclusion) sum-free set of size at most 2 and all of size 3 where there exists $a \in S$ such that $a \notin \langle S \setminus \{a\}\rangle$.

The commuting graph ${\cal C}(G,X)$, where $G$ is a group and $X$ a subset of $G$, has $X$ as its vertex set with two distinct elements of $X$ joined by an edge when they commute in $G$. Here the diameter and disc structure of ${\cal C}(G,X)$ is investigated when $G$ is the symmetric group and $X$ a conjugacy class of $G$.

In this article we consider commuting graphs of involution conjugacy classes in the affine Weyl group of typeC n . We show that where the graph is connected the diameter is at most n + 2. MSC(2000): 20F55, 05C25, 20D60. *

Abstract. Let W be the set of strongly real elements of a Coxeter group W . Then for w 2 W , the excess of w is defined byWhen W is finite we may also define E.w/, the reflection excess of w. The main result established here is that if W is finite and X is a W -conjugacy class, then there exists w 2 X such that w has minimal length in X and e.w/ D 0 D E.w/.