On the minimum degree of power graphs of finite nilpotent groups

Abstract

The power graph $\mathcal{P}(G)$ of a group G is the simple graph with vertex set G and two distinct vertices are adjacent if one of them is a positive power of the other. 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 $\left|G\right|$ 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 the degree of one of them. As an application, we obtain $\delta (\mathcal{P}(G))$ for some classes of nilpotent groups G.

Keywords:

• Euler’s totient function
• minimum degree
• nilpotent group
• power graph

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D15
• 05C25
• 05C07

Disclosure statement

No potential conflict of interest was reported by the author(s).