# Vertex connectivity of the power graph of a finite cyclic group

## TL;DR

This paper investigates the vertex connectivity of the power graph of finite cyclic groups, providing exact values for certain cases and bounds for others, enhancing understanding of the graph's structural properties.

## Contribution

It determines the exact vertex connectivity of power graphs for cyclic groups with multiple prime factors under specific conditions, and offers bounds in more general cases.

## Key findings

- Exact vertex connectivity for r=1 case.
- Exact values for certain composite orders with multiple primes.
- Upper bounds for general cases, sharp in many instances.

## Abstract

Let $n=p_1^{n_1}p_2^{n_2}\ldots p_r^{n_r}$, where $r,n_1,\ldots, n_r$ are positive integers and $p_1,p_2,\ldots,p_r$ are distinct prime numbers with $p_1<p_2<\cdots <p_r$. For the cyclic group $C_n$ of order $n$, let $\mathcal{P}(C_n)$ be the power graph of $C_n$ and $\kappa(\mathcal{P}(C_n))$ be the vertex connectivity of $\mathcal{P}(C_n)$. It is known that $\kappa(\mathcal{P}(C_n))=p_1^{n_1} -1$ if $r=1$. For $r\geq 2$, we determine the exact value of $\kappa(\mathcal{P}(C_n))$ when $2\phi(p_1\ldots p_{r-1})\geq p_1\ldots p_{r-1}$, and give an upper bound for $\kappa(\mathcal{P}(C_n))$ when $2\phi(p_1\ldots p_{r-1}) < p_1\ldots p_{r-1}$, which is sharp for many values of $n$ but equality need not hold always.