The Non-Commuting, Non-Generating Graph of a Nilpotent Group

  • Peter J. Cameron
  • Saul D. Freedman
  • Colva M. Roney-Dougal


For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.

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