On the Clique Numbers of Non-commuting Graphs of Certain Groups

Let G be a non-abelian group. The non-commuting graph A(G) of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Gamma, the maximum size of complete subgraphs of Gamma is called the clique number of Gamma and denoted by omega(Gamma). In this paper, we characterize all non-solvable groups G with omega(A(G)) <= 57 where 57 is the clique number of the non-commuting graph of the projective special linear group PSL(2,7). We also determine omega(A(G)) for all finite minimal simple groups G.