Super graphs on groups, I

G. Arunkumar; Peter J. Cameron; Rajat Kanti Nath; Lavanya Selvaganesh

doi:10.1007/s00373-022-02496-w

Super graphs on groups, I

G. Arunkumar, Peter J. Cameron^*, Rajat Kanti Nath, Lavanya Selvaganesh

^*Corresponding author for this work

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

Abstract

Let G be a finite group. A number of graphs with the vertex set G have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful to study them together. In addition, several authors have considered modifying the definition of these graphs by choosing a natural equivalence relation on the group such as equality, conjugacy, or equal orders, and joining two elements if there are elements in their equivalence class that are adjacent in the original graph. In this way, we enlarge the hierarchy into a second dimension. Using the three graph types and three equivalence relations mentioned gives nine graphs, of which in general only two coincide; we find conditions on the group for some other pairs to be equal. These often define interesting classes of groups, such as EPPO groups, 2-Engel groups, and Dedekind groups.

We study some properties of graphs in this new hierarchy. In particular, we
characterize the groups for which the graphs are complete, and
in most cases, we characterize the dominant vertices (those joined to all
others). Also, we give some results about universality, perfectness, and clique
number.

Original language	English
Article number	100
Number of pages	14
Journal	Graphs and Combinatorics
Volume	38
Issue number	3
Early online date	23 May 2022
DOIs	https://doi.org/10.1007/s00373-022-02496-w
Publication status	Published - Jun 2022

Keywords

Power graph
Commuting graph
Conjugacy
2-Engel groups
EPPO groups

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Cite this

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title = "Super graphs on groups, I",

abstract = "Let G be a finite group. A number of graphs with the vertex set G have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful to study them together. In addition, several authors have considered modifying the definition of these graphs by choosing a natural equivalence relation on the group such as equality, conjugacy, or equal orders, and joining two elements if there are elements in their equivalence class that are adjacent in the original graph. In this way, we enlarge the hierarchy into a second dimension. Using the three graph types and three equivalence relations mentioned gives nine graphs, of which in general only two coincide; we find conditions on the group for some other pairs to be equal. These often define interesting classes of groups, such as EPPO groups, 2-Engel groups, and Dedekind groups.We study some properties of graphs in this new hierarchy. In particular, wecharacterize the groups for which the graphs are complete, andin most cases, we characterize the dominant vertices (those joined to allothers). Also, we give some results about universality, perfectness, and cliquenumber.",

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author = "G. Arunkumar and Cameron, {Peter J.} and Nath, {Rajat Kanti} and Lavanya Selvaganesh",

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doi = "10.1007/s00373-022-02496-w",

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T1 - Super graphs on groups, I

AU - Arunkumar, G.

AU - Cameron, Peter J.

AU - Nath, Rajat Kanti

AU - Selvaganesh, Lavanya

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Y1 - 2022/6

N2 - Let G be a finite group. A number of graphs with the vertex set G have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful to study them together. In addition, several authors have considered modifying the definition of these graphs by choosing a natural equivalence relation on the group such as equality, conjugacy, or equal orders, and joining two elements if there are elements in their equivalence class that are adjacent in the original graph. In this way, we enlarge the hierarchy into a second dimension. Using the three graph types and three equivalence relations mentioned gives nine graphs, of which in general only two coincide; we find conditions on the group for some other pairs to be equal. These often define interesting classes of groups, such as EPPO groups, 2-Engel groups, and Dedekind groups.We study some properties of graphs in this new hierarchy. In particular, wecharacterize the groups for which the graphs are complete, andin most cases, we characterize the dominant vertices (those joined to allothers). Also, we give some results about universality, perfectness, and cliquenumber.

AB - Let G be a finite group. A number of graphs with the vertex set G have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful to study them together. In addition, several authors have considered modifying the definition of these graphs by choosing a natural equivalence relation on the group such as equality, conjugacy, or equal orders, and joining two elements if there are elements in their equivalence class that are adjacent in the original graph. In this way, we enlarge the hierarchy into a second dimension. Using the three graph types and three equivalence relations mentioned gives nine graphs, of which in general only two coincide; we find conditions on the group for some other pairs to be equal. These often define interesting classes of groups, such as EPPO groups, 2-Engel groups, and Dedekind groups.We study some properties of graphs in this new hierarchy. In particular, wecharacterize the groups for which the graphs are complete, andin most cases, we characterize the dominant vertices (those joined to allothers). Also, we give some results about universality, perfectness, and cliquenumber.

KW - Power graph

KW - Commuting graph

KW - Conjugacy

KW - 2-Engel groups

KW - EPPO groups

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DO - 10.1007/s00373-022-02496-w

M3 - Article

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VL - 38

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

IS - 3

M1 - 100

ER -

Super graphs on groups, I

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Keywords

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