Let G be a non-abelian group. The non-commuting graph AG of G is defned as the graph whose vertex set is the non-central elements of G and two vertices are jointif and only if they do not commute. In a ¯nite simple graph ¡, the maximum size ofcomplete subgraphs of ¡ is called the clique number of ¡ and denoted by !(¡). In thispaper, we characterize all non-solvable groups G with !(AG) · 57, where 57 is the clique number of the non-commuting graph of the projective special linear group PSL(2; 7). Wealso determine !(AG) for all ¯nite minimal simple groups G.
