Commuting Involution Graphs in Special Linear Groups

The diameter of the commuting graph of a finite group with trivial centre

“…The centralizer of a fundamental SL 2 (r a ) is non-trivial unless K ∼ = PSL 3 (2) or K ∼ = PSL 3 (4). Furthermore, if the root system of K is not A 2 , then a fundamental SL 2 (r a ) commutes with an element of order r.…”

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].…”