Line graph characterization of power graphs of finite nilpotent groups

TL;DR

This paper classifies finite nilpotent groups based on whether their power and proper power graphs are line graphs, providing a detailed characterization and conditions for various classes of groups.

Contribution

It offers a complete classification of finite nilpotent groups with line graph power graphs and explores conditions for specific groups like quaternion and dihedral groups.

Findings

Finite nilpotent groups with line graph power graphs are fully classified.

Conditions are established for proper power graphs of quaternion and dihedral groups to be line graphs.

Specific group orders are identified where proper power graphs are line graphs.

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.