On the minimum cut-sets of the power graph of a finite cyclic group, II
Sanjay Mukherjee, Kamal Lochan Patra, Binod Kumar Sahoo
TL;DR
This work advances the understanding of the power graph $\mathcal{P}(C_n)$ of a finite cyclic group by precisely determining minimum cut-sets and the vertex connectivity $\kappa(\mathcal{P}(C_n))$ for new regimes with at least four distinct primes in $n$. Building on prior results for $r\le 3$ and the two-candidate framework from MPS, the authors introduce two cut-set families $Z_a^s$ and $X_{a,b}^{s,t}$ and accompanying upper bounds $\beta_a^s$ and $\theta_{a,b}^{s,t}$, then derive exact classifications in three main cases: (i) $r\ge4$, $n_r\ge2$ with a defined index set $\Omega$; (ii) $r\in\{4,5\}$, $n_r=1$, and $p_1\in\{2,3\}$; and (iii) explicit subcases when $r=4$ or $r=5$ with particular prime configurations. The results yield explicit formulas for $\kappa(\mathcal{P}(C_n))$, reveal when the extremal cut-sets are unique, and provide corollaries for important parameter regimes, thereby giving a near-complete picture of connectivity for these power graphs in the new parameter ranges.
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) $r=5$, $n_r=1$, $p_1\geq 3$.