Commuting involution graphs for sporadic simple groups

Abstract

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.

Involution Graphs Where the Product of Two Adjacent Vertices Has Order Three

“…It was seen in [11] that the graph (PSL(2, 11), X, S), with X a conjugacy class of involutions and S a conjugacy class of S 3 -subgroups, is isomorphic to its dual graph, with the duality between X and S induced by elements of PGL(2, 11) \ PSL (2,11). We now show that the only other value of q for which this happens is q = 13.…”

“…Let G be a group with a nonempty set X of involutions closed under conjugation and a nonempty set S of S 3 -subgroups also closed under conjugation. The S 3 -involution graph (G, X, S) of G with respect to X and S is the graph with vertices the elements of X such that two vertices x, y are adjacent if and only if x, y ∈ S. In order to avoid degeneracies, we always require that X is the set of all involutions contained in elements of S. The tower of graphs is then given by a series of S 3 -involution graphs for A 5 , PSL (2,11), M 11 and M 12 , where, for each group, both X and S are single conjugacy classes. The existence of this tower suggests the following natural problem.…”

“…If G is a 3-transposition group such that X is a class of 3-transpositions and S is the set of all S 3 -subgroups generated by pairs of elements of X , then C(G, X ) is the complement of (G, X, S). Commuting involution graphs for groups other than 3-transposition groups have recently been studied in [2][3][4].…”

On local fusion graphs of finite Coxeter groups

“…For example, commuting involution graphs have a conjugacy class of involutions as their vertex set, with distinct vertices joined by an edge if, and only if, the relevant involutions commute. In [4][5][6], properties of these graphs are investigated for a variety of groups, including finite Coxeter groups. Even more closely related to the graphs of the present article are S 3 -involution graphs, where vertices are again involutions, and adjacent vertices must have product order 3.…”

On Coprimality Graphs for Symmetric Groups

“…Such graphs, called local fusion graphs and denoted by F (G, X), have been investigated by the authors in [2] when G is a symmetric group and X is a conjugacy class of involutions (see Theorem 2.2 in Section 2). While C {2} (G, X) when X is a G-conjugacy class of involutions is a commuting involution graph -such graphs have been studied in [3], [4], [5], [6] and [7]. Also certain types of coprimality graph appear in [9].…”