## Abstract

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 the enhanced power graph.

For an arbitrary pair of these three graphs we characterize finite groups for which

this pair of graphs are equal.

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 the enhanced power graph.

For an arbitrary pair of these three graphs we characterize finite groups for which

this pair of graphs are equal.

Original language | English |
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Article number | 3.16 |

Number of pages | 18 |

Journal | Electronic Journal of Combinatorics |

Volume | 24 |

Issue number | 3 |

Publication status | Published - 27 Jul 2017 |

## Keywords

- power graph
- clique number
- group
- independence number
- chromatic number