Solvable conjugacy class graph of groups

Parthajit Bhowal; Peter J. Cameron; Rajat Kanti Nath; Benjamin Sambale

doi:10.1016/j.disc.2023.113467

Solvable conjugacy class graph of groups

Parthajit Bhowal
, Peter J. Cameron^*
, Rajat Kanti Nath
, Benjamin Sambale

^*Corresponding author for this work

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

Abstract

In this paper we introduce the graph Γ_sc(G) associated with a group G, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist x ∈ C and y ∈ D such that x and y generate a solvable group.

We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number d, and we find explicitly the list of such groups with d=2. We pose some problems on the relation of the SCC-graph to the solvable graph and to the NCC-graph, which we cannot solve.

Original language	English
Article number	113467
Number of pages	8
Journal	Discrete Mathematics
Volume	346
Issue number	8
Early online date	20 Apr 2023
DOIs	https://doi.org/10.1016/j.disc.2023.113467
Publication status	Published - 1 Aug 2023

Keywords

Clique number
Non-solvable group
Graph
Girth
Conjugacy class

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Bhowal_2023_DM_Solvable-conjugacy_CC
Copyright © 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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title = "Solvable conjugacy class graph of groups",

abstract = "In this paper we introduce the graph Γsc(G) associated with a group G, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist x ∈ C and y ∈ D such that x and y generate a solvable group. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number d, and we find explicitly the list of such groups with d=2. We pose some problems on the relation of the SCC-graph to the solvable graph and to the NCC-graph, which we cannot solve. ",

keywords = "Clique number, Non-solvable group, Graph, Girth, Conjugacy class",

author = "Parthajit Bhowal and Cameron, \{Peter J.\} and Nath, \{Rajat Kanti\} and Benjamin Sambale",

note = "Funding: The fourth author [BS] is supported by the German Research Foundation (SA 2864/1-2 and SA 2864/3-1).",

year = "2023",

month = aug,

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doi = "10.1016/j.disc.2023.113467",

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T1 - Solvable conjugacy class graph of groups

AU - Bhowal, Parthajit

AU - Cameron, Peter J.

AU - Nath, Rajat Kanti

AU - Sambale, Benjamin

N1 - Funding: The fourth author [BS] is supported by the German Research Foundation (SA 2864/1-2 and SA 2864/3-1).

PY - 2023/8/1

Y1 - 2023/8/1

N2 - In this paper we introduce the graph Γsc(G) associated with a group G, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist x ∈ C and y ∈ D such that x and y generate a solvable group. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number d, and we find explicitly the list of such groups with d=2. We pose some problems on the relation of the SCC-graph to the solvable graph and to the NCC-graph, which we cannot solve.

AB - In this paper we introduce the graph Γsc(G) associated with a group G, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist x ∈ C and y ∈ D such that x and y generate a solvable group. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number d, and we find explicitly the list of such groups with d=2. We pose some problems on the relation of the SCC-graph to the solvable graph and to the NCC-graph, which we cannot solve.

KW - Clique number

KW - Non-solvable group

KW - Graph

KW - Girth

KW - Conjugacy class

U2 - 10.1016/j.disc.2023.113467

DO - 10.1016/j.disc.2023.113467

M3 - Article

SN - 0012-365X

VL - 346

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 8

M1 - 113467

ER -

Solvable conjugacy class graph of groups

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Keywords

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