A4-Graph of Finite Simple Groups

  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.