Line graph characterization of power graphs of finite nilpotent groups

Abstract

This paper deals with the classification of groups G such that power graphs and proper power graphs of G are line graphs. In fact, we classify all finite nilpotent groups whose power graphs are line graphs. Also, we categorize all finite nilpotent groups (except non-abelian 2-groups) whose proper power graphs are line graphs. Moreover, we investigate when the proper power graphs of generalized quaternion groups are line graphs. Besides, we derive a condition on the order of the dihedral groups for which the proper power graphs of the dihedral groups are line graphs.

Keywords:

• Line graph
• nilpotent group
• power graph
• proper power graph

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 05C25
• 20D10

Acknowledgment

The author profusely thanks the anonymous referee for meticulous reading of the manuscript and valuable suggestions that significantly improved the exposition of this paper. The author would like to thank Prof. Arvind Ayyer for his constant support and encouragement. Also, the author would like to thank Dr. Sumana Hatui for the helpful discussions on p-groups.

Additional information

Funding

The author was supported by NBHM Post Doctoral Fellowship grant 0204/52/2019/RD-II/339.