Investigation of Commuting Graphs for Elements of Order 3 in Certain Leech Lattice Groups

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.

The Discs Structures of A4-Graph for the Held Group

“…Moreover, the action of C G (t) on 3A breaks into the discs of the A4-graph as X C sets for He-conjugacy class C. This can explain in the next table: In this case, we let X=3B and then |X|=7996800. Let t be a fixed in 3B, then C G (t) isomorphic to 3.PSL (3,2). The computational calculation yields that A 4 (He,3B) is connected with diameter 4.…”

“…Then the A4-graph is connected with diameter 3. The computational calculations inside GAP [8] led to the following results:  0 (t)={t}, 1 (t)={ (3,4,6), (1,4,6), (2,4,6), (3,5,4), (1,5,4), (2,5,4), (3,6,5), (1,6,5),(2,6,5)},  2 (t)={ (3,4,5), (1,4,5), (2,4,5), (3,5,6), (1,5,6), (2,5,6), (3,6,4), (1,6,4), (2,6,4),…”

“…A group action on a graph is one of the most effective ways for understanding group structure. Several recent research, including [1,2] and [3], have established the effectiveness of this strategy. For a finite group G, it contains a subset X as G-conjugacy of an element of order 3.…”

More on Result Involution Graphs

“…Then the commuting involution graph is introduced on 𝐺, whose vertex set a subset of involution elements in 𝐺 and two vertices are adjacent if they commute see [1,2]. Furthermore, in [3,4,5] the most popular interest work in this context can be seen. Moreover, involution graphs, which adjacency of two vertices is defined by their product is of order 3, are describe in full details in [6].…”