On the minimum degree of power graphs of finite nilpotent groups

TL;DR

This paper investigates the minimum degree of the power graph of finite noncyclic nilpotent groups, identifying key vertices that determine this degree and applying results to certain abelian groups.

Contribution

It characterizes the minimum degree of power graphs for finite noncyclic nilpotent groups, linking it to generators of maximal cyclic subgroups under specific conditions.

Findings

Identifies vertices with minimal degree related to maximal cyclic subgroups

Provides formulas for minimum degree in certain finite abelian groups

Establishes conditions involving Sylow subgroups and prime divisors

Abstract

The power graph $\mathcal{P}(G)$ of a group $G$ is the simple graph with vertex set $G$ and two vertices are adjacent whenever one of them is a positive power of the other. In this paper, for a finite noncyclic nilpotent group $G$, we study the minimum degree $\delta(\mathcal{P}(G))$ of $\mathcal{P}(G)$. Under some conditions involving the prime divisors of $|G|$ and the Sylow subgroups of $G$, we identify certain vertices associated with the generators of maximal cyclic subgroups of $G$ such that $\delta(\mathcal{P}(G))$ is equal to the degree of one of these vertices. As an application, we obtain $\delta(\mathcal{P}(G))$ for some classes of finite noncyclic abelian groups $G$.