On Finite groups whose power graphs are line graphs

TL;DR

This paper extends the characterization of finite groups with line graph power graphs from nilpotent groups to all finite groups, correcting previous results and classifying groups based on various power graph properties.

Contribution

It generalizes existing results to all finite groups, corrects previous classifications for dihedral groups, and classifies groups based on enhanced power graph line graph properties.

Findings

Classified all finite groups with power graphs as line graphs.

Corrected previous results for dihedral groups' proper power graphs.

Identified groups whose various power graphs are complements of line graphs.

Abstract

S. Bera (Line graph characterization of power graphs of finite nilpotent groups, \textit{Communication in Algebra}, 50(11), 4652-4668, 2022) characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs. In this paper, we extend the results of above mentioned paper to arbitrary finite groups. Also, we correct the corresponding result of the proper power graphs of dihedral groups. Moreover, we classify all the finite groups whose enhanced power graphs are line graphs. We classify all the finite nilpotent groups (except non-abelian $2$-groups) whose proper enhanced power graphs are line graphs of some graphs. Finally, we determine all the finite groups whose power graphs, proper power graphs, enhanced power graphs and proper enhanced power graphs are the complement of line graphs, respectively.