Domination number of the non-commuting graph of finite groups - Ebrahim Vatandoost; Masoumeh Khalili (Electronic Journal of Graph Theory and Applications (EJGTA), 2018)
Let G be a non-abelian group. The non-commuting graph of group G, shown by ΓG, 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 ab ≠ ba. 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 $γ(Γ_G)<\frac{|G|-|Z(G)|}{2}.$ Also we charactrize all of groups G of order n with t = ∣Z(G)∣, in which $γ(Γ_{G})+γ(\overline{Γ}_{G})\in \{n-t+1,n-t,n-t-1,n-t-2\}.$