On neighborhoods in the enhanced power graph associated with a finite group

TL;DR

This paper studies the structure of neighborhoods in the enhanced power graph of finite groups, focusing on characterizing p-groups with minimal maximum neighborhood sizes for nontrivial elements.

Contribution

It provides a characterization of finite p-groups with the smallest maximum neighborhood size in their enhanced power graph.

Findings

Identifies p-groups with minimal maximum neighborhood sizes

Provides structural insights into enhanced power graphs of finite groups

Characterizes neighborhoods in the cyclic graph context

Abstract

This article investigates neighborhoods' sizes in the enhanced power graph (as known as the cyclic graph) associated with a finite group. In particular, we characterize finite $p$-groups with the smallest maximum size for neighborhoods of nontrivial element in its enhanced power graph.