On the difference of the enhanced power graph and the power graph of a finite group

Sucharita Biswas; Peter J. Cameron; Angsuman Das; Hiranya Kishore Dey

doi:10.1016/j.jcta.2024.105932

On the difference of the enhanced power graph and the power graph of a finite group

Sucharita Biswas
, Peter J. Cameron
, Angsuman Das^*
, Hiranya Kishore Dey

^*Corresponding author for this work

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

Abstract

The difference graph of a finite group D (G) is the difference of the enhanced power graph of G and the power graph of G, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of D (G) with respect to various properties of the underlying group G. We also find several connections between the difference graph of G and the Gruenberg-Kegel graph of G. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth.

Original language	English
Article number	105932
Number of pages	31
Journal	Journal of Combinatorial Theory, Series A
Volume	208
Early online date	21 Jun 2024
DOIs	https://doi.org/10.1016/j.jcta.2024.105932
Publication status	Published - Nov 2024

Keywords

Power graph
Enhanced power graph
Twin reduction
Gruenberg-Kegel graph (prime graph)

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10.1016/j.jcta.2024.105932

Biswas-2024-On-the-difference-of-JCTA-AAM-CCBY
Copyright © 2024 the Authors. This work has been made available online in accordance with the University of St Andrews Open Access policy. This accepted manuscript is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The final published version of this work is available at https://doi.org/10.1016/j.jcta.2024.105932.
Accepted author manuscript, 403 KBLicence: CC BY

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title = "On the difference of the enhanced power graph and the power graph of a finite group",

abstract = "The difference graph of a finite group D (G) is the difference of the enhanced power graph of G and the power graph of G, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of D (G) with respect to various properties of the underlying group G. We also find several connections between the difference graph of G and the Gruenberg-Kegel graph of G. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth.",

keywords = "Power graph, Enhanced power graph, Twin reduction, Gruenberg-Kegel graph (prime graph)",

author = "Sucharita Biswas and Cameron, \{Peter J.\} and Angsuman Das and Dey, \{Hiranya Kishore\}",

note = "Funding: The first author is supported by the PhD fellowship of CSIR (File no. 08/155 (0086)/2020 − EMR − I), Govt. of India. The second author acknowledges the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no. EP/R014604/1), where he held a Simons Fellowship. The third author acknowledges the funding of DST grant SR/F ST/MS − I/2019/41 and MT R/2022/000020, Govt. of India. The fourth author acknowledges SERB-National Post-Doctoral Fellowship (File No. PDF/2021/001899) during the preparation of this work. ",

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doi = "10.1016/j.jcta.2024.105932",

language = "English",

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AU - Cameron, Peter J.

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AU - Dey, Hiranya Kishore

N1 - Funding: The first author is supported by the PhD fellowship of CSIR (File no. 08/155 (0086)/2020 − EMR − I), Govt. of India. The second author acknowledges the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no. EP/R014604/1), where he held a Simons Fellowship. The third author acknowledges the funding of DST grant SR/F ST/MS − I/2019/41 and MT R/2022/000020, Govt. of India. The fourth author acknowledges SERB-National Post-Doctoral Fellowship (File No. PDF/2021/001899) during the preparation of this work.

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N2 - The difference graph of a finite group D (G) is the difference of the enhanced power graph of G and the power graph of G, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of D (G) with respect to various properties of the underlying group G. We also find several connections between the difference graph of G and the Gruenberg-Kegel graph of G. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth.

AB - The difference graph of a finite group D (G) is the difference of the enhanced power graph of G and the power graph of G, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of D (G) with respect to various properties of the underlying group G. We also find several connections between the difference graph of G and the Gruenberg-Kegel graph of G. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth.

KW - Power graph

KW - Enhanced power graph

KW - Twin reduction

KW - Gruenberg-Kegel graph (prime graph)

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DO - 10.1016/j.jcta.2024.105932

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JF - Journal of Combinatorial Theory, Series A

M1 - 105932

ER -

On the difference of the enhanced power graph and the power graph of a finite group

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Keywords

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