Domination number of the non-commuting graph of finite groups

Ebrahim Vatandoost (Department of Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran)
Masoumeh Khalili (Department of Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 6, No 2 (2018): Electronic Journal of Graph Theory and Applications

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. </p><p>Here, we study some properties of the non-commuting graph of some finite groups. In this paper, we show that $\gamma(\Gamma_G)&lt;\frac{|G|-|Z(G)|}{2}.$ Also we charactrize all of groups G of order n with t = ∣Z(G)∣, in which $\gamma(\Gamma_{G})+\gamma(\overline{\Gamma}_{G})\in \{n-t+1,n-t,n-t-1,n-t-2\}.$