On the clique numbers of non-commuting graphs of certain groups

Abstract

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 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 ω(AG)≤ 57, where 57 is the clique number of the non-commuting graph of the projective special linear group PSL(2,7). We also determine ω(AG) for all finite minimal simple groups G. © 2010 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.