Between the enhanced power graph and the commuting graph

Peter J. Cameron; Bojan Kuzma

doi:10.1002/jgt.22871

Between the enhanced power graph and the commuting graph

Peter J. Cameron^*, Bojan Kuzma

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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 automorphisms of G act as automorphisms of the deep commuting graph.

Original language	English
Pages (from-to)	295-303
Number of pages	9
Journal	Journal of Graph Theory
Volume	102
Issue number	2
Early online date	8 Aug 2022
DOIs	https://doi.org/10.1002/jgt.22871
Publication status	Published - Feb 2023

Keywords

Commuting graph
Enhanced power graph
Central extension
Schur multiplier
Bogomolov multiplier

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Cameron-2022-Between-the-enhanced-power-graph-JGraphTheory-22871-CC
Copyright © 2022 The Authors. Journal of Graph Theory published by Wiley Periodicals LLC. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
Final published version, 393 KBLicence: CC BY-NC-ND

Cite this

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title = "Between the enhanced power graph and the commuting graph",

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 automorphisms of G act as automorphisms of the deep commuting graph.",

keywords = "Commuting graph, Enhanced power graph, Central extension, Schur multiplier, Bogomolov multiplier",

author = "Cameron, {Peter J.} and Bojan Kuzma",

note = "Funding: The second author acknowledges the financial support from the Slovenian Research Agency, ARRS (research core funding No. P1-0222, No. P1-0285, and research project No. N1-0210).",

year = "2023",

month = feb,

doi = "10.1002/jgt.22871",

language = "English",

volume = "102",

pages = "295--303",

journal = "Journal of Graph Theory",

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T1 - Between the enhanced power graph and the commuting graph

AU - Cameron, Peter J.

AU - Kuzma, Bojan

N1 - Funding: The second author acknowledges the financial support from the Slovenian Research Agency, ARRS (research core funding No. P1-0222, No. P1-0285, and research project No. N1-0210).

PY - 2023/2

Y1 - 2023/2

N2 - 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 automorphisms of G act as automorphisms of the deep commuting graph.

AB - 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 automorphisms of G act as automorphisms of the deep commuting graph.

KW - Commuting graph

KW - Enhanced power graph

KW - Central extension

KW - Schur multiplier

KW - Bogomolov multiplier

U2 - 10.1002/jgt.22871

DO - 10.1002/jgt.22871

M3 - Article

SN - 0364-9024

VL - 102

SP - 295

EP - 303

JO - Journal of Graph Theory

JF - Journal of Graph Theory

IS - 2

ER -

Between the enhanced power graph and the commuting graph

Abstract

Keywords

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