Vertex connectivity of the power graph of a finite cyclic group

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.