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

Let G be a non-abelian group. The non-commuting graph AG 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 Γ the maximum size of a complete subgraph of Γ is called the clique number of Γ and it is denoted by ω(Γ). In this paper we characterize all non-solvable groups G with ω(AG) ≤ 57, where the number 57 is the clique number of the non-commuting graph of the projective special linear group PSL(2, 7). We also complete the determination of ω(AG) for all finite minimal simple groups.