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.
@article{10_37236_9802,
author = {Peter Cameron and Saul Freedman and Colva Roney-Dougal},
title = {The non-commuting, non-generating graph of a nilpotent group},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {1},
doi = {10.37236/9802},
zbl = {1456.05072},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9802/}
}
TY - JOUR
AU - Peter Cameron
AU - Saul Freedman
AU - Colva Roney-Dougal
TI - The non-commuting, non-generating graph of a nilpotent group
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/9802/
DO - 10.37236/9802
ID - 10_37236_9802
ER -
%0 Journal Article
%A Peter Cameron
%A Saul Freedman
%A Colva Roney-Dougal
%T The non-commuting, non-generating graph of a nilpotent group
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/9802/
%R 10.37236/9802
%F 10_37236_9802
Peter Cameron; Saul Freedman; Colva Roney-Dougal. The non-commuting, non-generating graph of a nilpotent group. The electronic journal of combinatorics, Tome 28 (2021) no. 1. doi: 10.37236/9802
