On the Connectivity and Independence Number of Power Graphs of Groups

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 indepen...

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Bibliographic Details
Published in:Graphs and combinatorics Vol. 36; no. 3; pp. 895 - 904
Main Authors: Cameron, Peter J., Jafari, Sayyed Heidar
Format: Journal Article
Language:English
Published: Tokyo Springer Japan 01.05.2020
Springer Nature B.V
Subjects:
Combinatorics
Engineering Design
Graph theory
Graphs
Mathematics
Mathematics and Statistics
Original Paper
Connectivity
20D10
Power graph
Cyclic group
05C25
Independence number
ISSN:0911-0119, 1435-5914
Online Access:Get full text
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Summary: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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ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-020-02162-z