- - - Middle Technical University, Technical Instructors Training Institute, Iraq
- - - Middle Technical University, Technical Instructors Training Institute, Iraq
- - - Middle Technical University, Institute of Technology, Baghdad, Iraq

Assume that X is a subset of the finite group G. The A۴-graph is known as a simple graph denoted by \mathcal{A}_۴ (G, X) having X as a vertex set and two vertices x, y \in X, is linked by an edges if x\neq y and {xy}^{-۱} = {yx}^{-۱} . In this paper, we consider \mathcal{A}_۴ (G, X) when G is either Tits group T or Mathieu group M_{۲۰} 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 A۴-graph.

Finite simple groups, A۴-graph, connectivity, cliques