Non-nilpotent graph of a group

We associate a graph ${\mathcal{N}}_G$ with a group $G$ (called the non-nilpotent graph of $G$) as follows: take $G$ as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of ${\mathcal{N}}_G$ and its induced subgraph on $G\mathrm{\backslash }nil(G)$, where $nil(G)=\{x\in G\mathrm{\mid }<x,y>\text{is nilpotent for all}y\in G\}$. For any finite group $G$, we prove that ${\mathcal{N}}_G$ has either $\mathrm{\mid }{Z}^{\ast }(G)\mathrm{\mid }$ or $\mathrm{\mid }{Z}^{\ast }(G)\mathrm{\mid }+1$ connected components, where $Z^{\ast }(G)$ is the hypercenter of $G$. We give a new characterization for finite nilpotent groups in terms of the non-nilpotent graph. In fact we prove that a finite group $G$ is nilpotent if and only if the set of vertex degrees of ${\mathcal{N}}_G$ has at most two elements.