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

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.