Generalizing the enhanced power graph of a group with respect to automorphisms

TL;DR

This paper generalizes the enhanced power graph of a group by considering automorphism classes, analyzing its connectivity, diameter, universal vertices, and conditions for completeness or emptiness.

Contribution

It introduces a new automorphism-based generalization of the enhanced power graph and studies its fundamental properties and classifications.

Findings

The generalized graph has similar connectivity and diameter to the original.

Universal vertices are characterized within the generalized graph.

Conditions for the graph to be complete or empty are established.

Abstract

We generalize the enhanced power graph by replacing elements with classes under automorphisms. We show that the connectivity and diameter of this graph is similar to that of the enhanced power graph. We consider the universal vertices of this graph and when this graph is a complete graph. Finally, we classify when this graph is the empty graph.