On the Structure of the Power Graph and the Enhanced Power Graph of a Group

Author:

Aalipour Ghodratollah,Akbari Saieed,Cameron Peter J.,Nikandish Reza,Shaveisi Farzad

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 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.

Publisher

The Electronic Journal of Combinatorics

Subject

Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics

Cited by 42 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Nilpotent groups whose difference graphs have positive genus;Ricerche di Matematica;2023-12-02

2. On the difference graph of power graphs of finite groups;Quaestiones Mathematicae;2023-11-30

3. The non-commuting, non-generating graph of a non-simple group;Algebraic Combinatorics;2023-11-07

4. On groups with chordal power graph, including a classification in the case of finite simple groups;Journal of Algebraic Combinatorics;2023-10-04

5. Power graphs of all nilpotent groups;Journal of Algebraic Combinatorics;2023-09-22

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