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

Abstract

Let G be a non-abelian group. The non-commuting graph [Formula: see text] 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 [Formula: see text], where 57 is the clique number of the non-commuting graph of the projective special linear group PSL (2,7). We also determine [Formula: see text] for all finite minimal simple groups G.