Domination number of the non-commuting graph of finite groups

Let G be a non-abelian group. The non-commuting graph of group G, shown by Gamma(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 ab not equal ba. A set S subset of V (of vertices in a graph Gamma is a dominating set if every vertex v is an element of V (Gamma) is an element of S or adjacent to an element of S. The domination number of a graph Gamma denoted by gamma(Gamma), is the minimum size of a dominating set of Gamma. Here, we study some properties of the non-commuting graph of some finite groups. In this paper, we show that gamma(Gamma(G)) < vertical bar G vertical bar-vertical bar Z(G)vertical bar/2 Also we charactrize all of groups G of order n with t = vertical bar Z(G)vertical bar, in which gamma(Gamma(G)) + gamma(<(Gamma)over bar>(G)) is an element of{n - t + 1, n - t, n - t - 1, n - t - 2}.