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

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2018.6.2.3

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