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

TL;DR

This work introduces the automorphic enhanced power graph $\Delta(G,A)$, generalizing the enhanced power graph by using automorphism orbits and connecting two $A$-conjugacy classes whenever representatives generate a cyclic subgroup. It establishes connectivity and diameter bounds that parallel known results for the enhanced power graph, providing sharp diameter estimates in both non-$p$-group and $p$-group settings, along with a refinement for nilpotent groups. The authors develop a structural theory of universal vertices via $\mathrm{Cyc}_{G}(x)$ and $K(G)$, characterizing when $\Delta(G,A)$ has a universal vertex or is complete, including a precise treatment when $A\le\mathrm{Inn}(G)$. They also classify when $\Delta(G,A)$ is an empty graph, giving explicit group-theoretic conditions (including $A_{5}$-related cases) and contrasting with the Inn-only case. Overall, the paper extends the landscape of graph-Group interactions by linking automorphism-actions to graph connectivity, diameter, and extremal structures such as universal vertices, complete graphs, and empty graphs.

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.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 2.1
• Lemma 2.2