Regularity of extended conjugate graphs of finite groups

Piyapat Dangpat; Teerapong Suksumran; Piyapat Dangpat; Teerapong Suksumran

doi:10.3934/math.2022304

AIMS Mathematics

2022, Volume 7, Issue 4: 5480-5498. doi: 10.3934/math.2022304

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Research article

Regularity of extended conjugate graphs of finite groups

Piyapat Dangpat ¹,
Teerapong Suksumran ^{2
,
,}

1.
Master Program in Mathematics, Graduate School, Chiang Mai University, Chiang Mai 50200, Thailand
2.
Research Group in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received: 27 September 2021 Revised: 31 December 2021 Accepted: 04 January 2022 Published: 07 January 2022
MSC : Primary 20E45; Secondary 05C25, 20D99

The extended conjugate graph associated to a finite group $ G $ is defined as an undirected graph with vertex set $ G $ such that two distinct vertices joined by an edge if they are conjugate. In this article, we show that several properties of finite groups can be expressed in terms of properties of their extended conjugate graphs. In particular, we show that there is a strong connection between a graph-theoretic property, namely regularity, and an algebraic property, namely nilpotency. We then give some sufficient conditions and necessary conditions for the non-central part of an extended conjugate graph to be regular. Finally, we study extended conjugate graphs associated to groups of order $ pq $, $ p^3 $, and $ p^4 $, where $ p $ and $ q $ are distinct primes.
- extended conjugate graph,
- regular graph,
- conjugacy class,
- nilpotent group,
- $ p $-group
Citation: Piyapat Dangpat, Teerapong Suksumran. Regularity of extended conjugate graphs of finite groups[J]. AIMS Mathematics, 2022, 7(4): 5480-5498. doi: 10.3934/math.2022304

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Abstract

The extended conjugate graph associated to a finite group $ G $ is defined as an undirected graph with vertex set $ G $ such that two distinct vertices joined by an edge if they are conjugate. In this article, we show that several properties of finite groups can be expressed in terms of properties of their extended conjugate graphs. In particular, we show that there is a strong connection between a graph-theoretic property, namely regularity, and an algebraic property, namely nilpotency. We then give some sufficient conditions and necessary conditions for the non-central part of an extended conjugate graph to be regular. Finally, we study extended conjugate graphs associated to groups of order $ pq $, $ p^3 $, and $ p^4 $, where $ p $ and $ q $ are distinct primes.

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