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

TL;DR

This paper investigates the vertex connectivity of the power graph of finite cyclic groups, providing new bounds and exact values for cases with multiple prime factors, extending previous results.

Contribution

It introduces a new upper bound for the vertex connectivity of power graphs of cyclic groups with at least four prime factors and determines exact values when the largest prime exponent is at least two.

Findings

Established a new upper bound for vertex connectivity when r ≥ 4.

Determined exact vertex connectivity for cyclic groups where the largest prime exponent n_r ≥ 2.

Calculated vertex connectivity for cyclic groups with distinct prime factors.

Abstract

The power graph $\mathcal{P}(G)$ of a given finite group $G$ is the simple undirected graph whose vertices are the elements of $G$, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity $\kappa(\mathcal{P}(G))$ of $\mathcal{P}(G)$ is the minimum number of vertices which need to be removed from $G$ so that the induced subgraph of $\mathcal{P}(G)$ on the remaining vertices is disconnected or has only one vertex. For a positive integer $n$, let $C_n$ be the cyclic group of order $n$. Suppose that the prime power decomposition of $n$ is given by $n =p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, where $r\geq 1$, $n_1,n_2,\ldots, n_r$ are positive integers and $p_1,p_2,\ldots,p_r$ are prime numbers with $p_1<p_2<\cdots <p_r$. The vertex connectivity $\kappa(\mathcal{P}(C_n))$ of $\mathcal{P}(C_n)$ is known for $r\leq 3$, see \cite{panda, cps}. In this paper, for $r\geq 4$, we give a new upper bound for $\kappa(\mathcal{P}(C_n))$ and determine $\kappa(\mathcal{P}(C_n))$ when $n_r\geq 2$. We also determine $\kappa(\mathcal{P}(C_n))$ when $n$ is a product of distinct prime numbers.