Domination number of the non-commuting graph of finite groups

Abstract

Let G be a non-abelian group. The non-commuting graph of group G , shown by Γ G , is a graph with the vertex set G Z ( G ) , where Z ( G ) is the center of group G . Also two distinct vertices of a and b are adjacent whenever a b ≠ b a . A set S ⊆ V (Γ) of vertices in a graph Γ is a dominating set if every vertex v ∈ V (Γ) is an element of S or adjacent to an element of S . The domination number of a graph Γ denoted by γ (Γ) , is the minimum size of a dominating set of Γ . Here, we study some properties of the non-commuting graph of some finite groups. In this paper, we show that $\gamma(\Gamma_G)l\frac{|G|-|Z(G)|}{2}.$ Also we charactrize all of groups G of order n with t = ∣ Z ( G )∣ , in which $\gamma(\Gamma_{G})+\gamma(\overline{\Gamma}_{G})\in \{n-t+1,n-t,n-t-1,n-t-2\}.$