Let G be a finite group and X be a conjugacy class of order 3 in G. In this paper, we introduce a new type of graphs, namely A4-graph of  G, as a simple graph denoted by A4(G,X) which has X as a vertex set. Two vertices,  x and y, are adjacent if and only if  x≠y and  x y-1=      y x-1. General properties  of the A4-graph as well as the structure of A4(G,X) when G@ 3D4(2) will be studied.

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.

In the case of a finite simple group , and -conjugacy class of element of order 3, The A4-graph is define as simple graph denoted by A4 has as vertex set and are adjacent if and only if x≠y and xy-1 = yx-1.We aim to investigate computationally the structure of theA4 when Leech Lattice groups.

Suppose that G is a finite group and X is a G-conjugacy classes of involutions. The commuting involution graph $${\mathcal {C}}(G,X)$$ C ( G , X ) is the graph whose vertex set is X with $$x, y \in X$$ x , y ∈ X being joined if $$x \ne y$$ x ≠ y and $$xy = yx$$ x y = y x . Here for various exceptional Lie type groups of characteristic two we investigate their commuting involution graphs.

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