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

Let G be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ 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 Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with $\omega(\mathcal{A}_G)\leq 57$, where 57 is the clique number of the non-commuting graph of the projective special linear group PSL(2,7). We also determine $\omega(\mathcal{A}_G)$ for all finite minimal simple groups G. [ABSTRACT FROM AUTHOR]