On the minimum cut-sets of the power graph of a finite cyclic group, II

TL;DR

This paper characterizes the minimum cut-sets and vertex connectivity of the power graph of finite cyclic groups with multiple prime factors, extending previous results to cases where the number of prime factors is four or five.

Contribution

It explicitly determines the minimum cut-sets and vertex connectivity of the power graph of cyclic groups for cases with four or more prime factors, generalizing earlier work.

Findings

Explicit minimum cut-sets for r ≥ 4 cases

Vertex connectivity of power graphs for specific prime factor conditions

Extension of previous characterizations to more complex cyclic groups

Abstract

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$ and two distinct vertices are adjacent if one of them is a power of the other. Let $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r},$ where $p_1,p_2,\ldots,p_r$ are primes with $p_1<p_2<\cdots <p_r$ and $n_1,n_2,\ldots, n_r$ are positive integers. For the cyclic group $C_n$ of order $n$, the minimum cut-sets of $\mathcal{P}(C_n)$ are characterized in \cite{cps} for $r\leq 3$. Recently, in \cite{MPS}, certain cut-sets of $\mathcal{P}(C_n)$ are identified such that any minimum cut-set of $\mathcal{P}(C_n)$ must be one of them. In this paper, for $r\geq 4$, we explicitly determine the minimum cut-sets, in particular, the vertex connectivity of $\mathcal{P}(C_n)$ when: (i) $n_r\geq 2$, (ii) $r=4$ and $n_r=1$, and (iii) $r=5$, $n_r=1$, $p_1\geq 3$.