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arXiv · 2012.03789

Between the enhanced power graph and the commuting graph

Peter J. Cameron·Bojan Kuzma
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Abstract

The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting graph of $G$. Two elements of $G$ are joined in the deep commuting graph if and only if their inverse images in every central extension of $G$ commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that the automorphism group of $G$ acts as automorphisms of the deep commuting graph.

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Peter J. Cameron, Bojan Kuzma. 2020-12-07. Between the enhanced power graph and the commuting graph. https://arxiv.org/abs/2012.03789

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