Domination number of the non-commuting graph of finite groups - Ebrahim Vatandoost; Masoumeh Khalili (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 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)<|G|-|Z(G)|2. Also we charactrize all of groups G of order n with t = ∣Z(G)∣, in which γ(Γ_G)+γ(Γ_G){n-t+1,n-t,n-t-1,n-t-2}.