# On the minimum degree of the power graph of a finite cyclic group

## TL;DR

This paper investigates the minimum degree of the power graph of finite cyclic groups, providing new results for cases with three or more prime divisors, including explicit vertex identification under certain conditions.

## Contribution

It extends the understanding of the minimum degree of power graphs for cyclic groups with multiple prime divisors, identifying specific vertices related to this degree.

## Key findings

- Minimum degree known for groups with one or two prime divisors.
- Identification of vertices for minimum degree when r ≥ 3 under certain conditions.
- Explicit identification of two vertices for r=3 or when n is a product of distinct primes.

## Abstract

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. The minimum degree $\delta(\mathcal{P}(C_n))$ of $\mathcal{P}(C_n)$ is known for $r\in\{1,2\}$, see [18]. For $r\geq 3$, under certain conditions involving the prime divisors of $n$, we identify at most $r-1$ vertices such that $\delta(\mathcal{P}(C_n))$ is equal to the degree of at least one of these vertices. If $r=3$ or if $n$ is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of $n$.