Abstract
Assume that G is a finite group and X is a subset of G. The commuting graph is denoted by С(G,X) and has a set of vertices X with two distinct vertices x, y Î X, being connected together on the condition of xy = yx. In this paper, we investigate the structure of Ϲ(G,X) when G is a particular type of Leech lattice groups, namely Higman–Sims group HS and Janko group J2, along with X as a G-conjugacy class of elements of order 3. We will pay particular attention to analyze the discs’ structure and determinate the diameters, girths, and clique number for these graphs.
Highlights
Introduction and PreliminariesIt is believed that studying the action of a group on a graph is one of the best comprehensible ways of analyzing the structure of the group
Suppose that G is a group and X is a subset of G; the commuting graph denoted by С(G,X) has the set of vertices X with two vertices x, y X, which are connected if x ≠y, where xy = yx
The commuting graphs were first illustrated by Fowler and Brauer in a seminal paper [1]
Summary
Introduction and PreliminariesIt is believed that studying the action of a group on a graph is one of the best comprehensible ways of analyzing the structure of the group. The commuting graphs were first illustrated by Fowler and Brauer in a seminal paper [1]. They were eminent for giving evidence of a prescribed isomorphism of an involution centralizer, where there is a limited number of non-abelian groups capable of containing it. These graphs were extremely vital for the works of the Margulis-Platanov conjecture [2], as the graphs mentioned in [1] have X = G\{1} where 1 is the identity element of G).
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