Commuting involution graphs for finite Coxeter groups

On local fusion graphs of finite Coxeter groups

“…Since the groups C.A n / have been tackled previously in [1], here we concentrate on the groups C.B n / and C.D n /. The Coxeter group C.B n / may be considered as the group of signed permutations of n objects (see [4] or [15]). Let Sym.n/ act on the set D ¹1; : : : ; nº, and define the i -th 'sign change' to be the element which sends i to i and fixes all other j 2 .…”

“…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.…”

“…For any given r; m 2 N, there exists a finite group G with conjugacy class of involutions X such that F .G; X / has exactly m connected components, each of which has diameter r. Theorems 1.3 and 1.4 contrast with many results concerning the diameter of graphs related to local fusion graphs. For example, in [5] it is shown that for finite symmetric groups the diameter of commuting involution graphs is at most 4, while in [4] it is proved that for any other finite irreducible Coxeter group the diameter of a commuting involution graph is at most 5. Also, in [6], we find analysis of the commuting involutions graphs of the majority of the sporadic simple groups, and it is shown that their diameters are at most 4.…”

On $$\pi $$ π -Product Involution Graphs in Symmetric Groups

“…The case when X = G\Z (G), first studied in [8], has been the focus of interest recently-see [9,13,14]. When X is taken to be a G-conjugacy class of involutions, we get the so-called commuting involution graph, the subject of a number of papers (see [1][2][3][4]12,15,17]). …”

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].…”