studying the a4-graphs for elements of order 3 in tits group t and mathieu group m20

Assume that $x$ is a subset of the finite group $g$. the a4-graph is known as a simple graph denoted by $ mathcal{a}_4 (g, x) $ having $x$ as a vertex set and two vertices $ x, y in x,$ is linked by an edges if $ xneq y $ and $ {xy}^{-1} = {yx}^{-1} $. in this paper, we consider $ mathcal{a}_4 (g, x) $ when $g$ is either tits group t or mathieu group $m_{20}$ and $x$ is $g$-conjugacy class of elements of order three. valuable results reached, for example, disc structure, girth, clique number, and diameters of the a4-graph.