On the connectivity and independence number of power graphs of groups

Peter J. Cameron; Sayyed Jafari

doi:10.1007/s00373-020-02162-z

On the connectivity and independence number of power graphs of groups

Peter J. Cameron, Sayyed Jafari

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

17 Citations (Scopus)

6 Downloads (Pure)

Abstract

Let G be a group. The power graph of G is a graph with vertex set G in which two distinct elements x,y are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence number, show that they have clique cover number equal to their independence number, and calculate this number. The proper power graph is the induced subgraph of the power graph on the set G-{1}. A group whose proper power graph is connected must be either a torsion group or a torsion-free group; we give characterizations of some groups whose proper power graphs are connected.

Original language	English
Pages (from-to)	895–904
Number of pages	10
Journal	Graphs and Combinatorics
Volume	36
Early online date	31 Mar 2020
DOIs	https://doi.org/10.1007/s00373-020-02162-z
Publication status	Published - May 2020

Keywords

Power graph
Connectivity
Independence number
Cyclic

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10.1007/s00373-020-02162-zLicence: CC BY

Cameron-Jafari2020_Article_OnTheConnectivityAndIndependen
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Cite this

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author = "Cameron, {Peter J.} and Sayyed Jafari",

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AU - Jafari, Sayyed

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N2 - Let G be a group. The power graph of G is a graph with vertex set G in which two distinct elements x,y are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence number, show that they have clique cover number equal to their independence number, and calculate this number. The proper power graph is the induced subgraph of the power graph on the set G-{1}. A group whose proper power graph is connected must be either a torsion group or a torsion-free group; we give characterizations of some groups whose proper power graphs are connected.

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KW - Power graph

KW - Connectivity

KW - Independence number

KW - Cyclic

U2 - 10.1007/s00373-020-02162-z

DO - 10.1007/s00373-020-02162-z

M3 - Article

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JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

ER -

On the connectivity and independence number of power graphs of groups

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Keywords

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