1School of Mathematical Science Rochester Institute of Technology (RIT) Rochester, NY 14623, USA. 2Department of Mathematical Sciences Sharif University of Technology Tehran, Iran. 3School of Mathematics and Statistics University of St Andrews St Andrews, Fife KY16 9SS, UK. 4Department of Basic Sciences Jundi-Shapur University of Technology Dezful, Iran. 5Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah, Iran
The electronic journal of combinatorics, Tome 24 (2017) no. 3
Let $G$ be a group. The power graph of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group $G$, the clique number of the power graph of $G$ is at most countably infinite. We also measure how close the power graph is to the commuting graph by introducing a new graph which lies in between. We call this new graph as the enhanced power graph. For an arbitrary pair of these three graphs we characterize finite groups for which this pair of graphs are equal.
Ghodratollah Aalipour
1
;
Saieed Akbari
2
;
Peter J. Cameron
3
;
Reza Nikandish
4
;
Farzad Shaveisi
5
1
School of Mathematical Science
Rochester Institute of Technology (RIT)
Rochester, NY 14623, USA.
2
Department of Mathematical Sciences
Sharif University of Technology
Tehran, Iran.
3
School of Mathematics and Statistics
University of St Andrews
St Andrews, Fife KY16 9SS, UK.
4
Department of Basic Sciences
Jundi-Shapur University of Technology
Dezful, Iran.
5
Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah, Iran
@article{10_37236_6497,
author = {Ghodratollah Aalipour and Saieed Akbari and Peter J. Cameron and Reza Nikandish and Farzad Shaveisi},
title = {On the structure of the power graph and the enhanced power graph of a group},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {3},
doi = {10.37236/6497},
zbl = {1369.05059},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6497/}
}
TY - JOUR
AU - Ghodratollah Aalipour
AU - Saieed Akbari
AU - Peter J. Cameron
AU - Reza Nikandish
AU - Farzad Shaveisi
TI - On the structure of the power graph and the enhanced power graph of a group
JO - The electronic journal of combinatorics
PY - 2017
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%J The electronic journal of combinatorics
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Ghodratollah Aalipour; Saieed Akbari; Peter J. Cameron; Reza Nikandish; Farzad Shaveisi. On the structure of the power graph and the enhanced power graph of a group. The electronic journal of combinatorics, Tome 24 (2017) no. 3. doi: 10.37236/6497
