Non-nilpotent graph of a group

Abstract

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 $G\backslash nil(G)$, where $nil(G)=\{x\in G | < x,y> \text{is nilpotent for all} y\in G\}$. For any finite group $G$, we prove that $\mathcal{N}_G$ has either $|Z^*(G)|$ or $|Z^*(G)|+1$ connected components, where $Z^*(G)$ 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.