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arXiv · 2108.12569

On the connectivity of the non-generating graph

Andrea Lucchini·Daniele Nemmi
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Abstract

Given a 2-generated finite group $G$, the non-generating graph of $G$ has as vertices the elements of $G$ and two vertices are adjacent if and only if they are distinct and do not generate $G$. We consider the graph $\Sigma(G)$ obtained from the non-generating graph of $G$ by deleting the universal vertices. We prove that if the derived subgroup of $G$ is not nilpotent, then this graph is connected, with diameter at most 5. Moreover we give a complete classification of the finite groups $G$ such that $\Sigma(G)$ is disconnected.

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Andrea Lucchini, Daniele Nemmi. 2021-08-28. On the connectivity of the non-generating graph. https://arxiv.org/abs/2108.12569

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