Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Sanjay Mukherjee, Kamal Lochan Patra, Binod Kumar Sahoo

TL;DR

The paper investigates the minimum cut-sets of the power graph $\mathcal{P}(C_n)$ of a finite cyclic group. It introduces two families of cut-sets, $Z_a^s$ and $X_{a,b}^{s,t}$, and leverages Euler’s totient function to compare their sizes and characterize separations in $\mathcal{P}(C_n)$. For $r\ge 4$, it proves that any minimum cut-set must be one of $Z_r^1$, $Z_a^{n_a}$ (with $n_a\ge 2$), or $X_{a,b}^{s,t}$, thereby completing the classification beyond the previously treated cases $r\le 3$. The approach combines detailed order-counts of elements in $C_n$, careful partitioning of prime divisors via a separation, and case analysis on the number of primitive-order components contained in the cut-set. This sharpens the understanding of connectivity and separations in power graphs of cyclic groups and provides concrete canonical cut-sets for $r\ge 4$.

Abstract

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$, in which two distinct vertices are adjacent if one of them is a power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. The minimum cut-sets of $\mathcal{P}(C_n)$ are characterized in \cite{cps} for $r\leq 3$. In this paper, for $r\geq 4$, we identify certain cut-sets of $\mathcal{P}(C_n)$ such that any minimum cut-set of $\mathcal{P}(C_n)$ must be one of them.

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

TL;DR

The paper investigates the minimum cut-sets of the power graph of a finite cyclic group. It introduces two families of cut-sets, and , and leverages Euler’s totient function to compare their sizes and characterize separations in . For , it proves that any minimum cut-set must be one of , (with ), or , thereby completing the classification beyond the previously treated cases . The approach combines detailed order-counts of elements in , careful partitioning of prime divisors via a separation, and case analysis on the number of primitive-order components contained in the cut-set. This sharpens the understanding of connectivity and separations in power graphs of cyclic groups and provides concrete canonical cut-sets for .

Abstract

The power graph of a finite group is the simple graph with vertex set , in which two distinct vertices are adjacent if one of them is a power of the other. For an integer , let denote the cyclic group of order and let be the number of distinct prime divisors of . The minimum cut-sets of are characterized in \cite{cps} for . In this paper, for , we identify certain cut-sets of such that any minimum cut-set of must be one of them.
Paper Structure (13 sections, 34 theorems, 47 equations)

This paper contains 13 sections, 34 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Power graphs
  3. Cut-sets of $\mathcal{P}(C_n)$
  4. Main result
  5. Euler's totient function
  6. The cut-sets $Z_a^s$ and $X_{a,b}^{s,t}$
  7. The cut-sets $Z_a^s$
  8. The cut-sets $X_{a,b}^{s,t}$
  9. Distribution of the sets $E_{\frac{n}{p_i}},$ $i\in [r]$
  10. Proof of the Main Theorem
  11. The case $|W|=0$
  12. The case $|W|=1$
  13. The case $|W|=2$

Key Result

Theorem 1.1

Let $r\geq 4$ and $X$ be a minimum cut-set of $\mathcal{P}(C_n)$. Then $X=Z_r^1$; or $X=Z_a^{n_a}$ for some $a\in [r]$ with $n_a\geq 2$; or $X=X_{a,b}^{s,t}$ for some $a,b\in [r]$ with $a\neq b$, $s\in [n_a]$ and $t\in [n_b]$.

Theorems & Definitions (59)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 49 more