Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The diameter and dominating sets of the difference graph of a nilpotent group

Xuanlong Ma, Samir Zahirović, Katarina Žigerović

TL;DR

This study introduces the difference graph $\mathcal{D}(G)$, defined as the edgewise difference of $\mathcal{P}_E(G)$ and $\mathcal{P}(G)$ with isolated vertices removed, for finite groups. It shows that for finite nilpotent groups, the diameter is bounded by $4$, and classifies nilpotent groups by diameter values up to $4$. The authors develop dominating-set constructions, notably $\mathcal{F}(G)$ built from maximal cyclic subgroups, and provide complete descriptions in the cyclic case regarding domination numbers and diameter depending on the factorization of the group order. The results refine and extend previous work on difference graphs, including corollaries for dihedral and quaternion groups, and yield a clear dichotomy of diameter values for non-cyclic nilpotent groups based on exponent and subgroup structure.

Abstract

Given a finite group $G$, the difference graph of $G$, denoted by $\mathcal{D}(G)$, is the difference of the enhanced power graph of $G$ and the power graph of $G$, with all isolated vertices removed. This paper mainly studies the dominating sets of the difference graph of a finite group. In particular, we prove that the diameter of the difference graph of a nilpotent group has an upper bound of $4$. Furthermore, we generalize and refine the result by Biswas et al. by classifying all nilpotent groups whose difference graph has diameter $k$, for each $k\le 4$.

The diameter and dominating sets of the difference graph of a nilpotent group

TL;DR

This study introduces the difference graph , defined as the edgewise difference of and with isolated vertices removed, for finite groups. It shows that for finite nilpotent groups, the diameter is bounded by , and classifies nilpotent groups by diameter values up to . The authors develop dominating-set constructions, notably built from maximal cyclic subgroups, and provide complete descriptions in the cyclic case regarding domination numbers and diameter depending on the factorization of the group order. The results refine and extend previous work on difference graphs, including corollaries for dihedral and quaternion groups, and yield a clear dichotomy of diameter values for non-cyclic nilpotent groups based on exponent and subgroup structure.

Abstract

Given a finite group , the difference graph of , denoted by , is the difference of the enhanced power graph of and the power graph of , with all isolated vertices removed. This paper mainly studies the dominating sets of the difference graph of a finite group. In particular, we prove that the diameter of the difference graph of a nilpotent group has an upper bound of . Furthermore, we generalize and refine the result by Biswas et al. by classifying all nilpotent groups whose difference graph has diameter , for each .
Paper Structure (5 sections, 18 theorems, 34 equations)

This paper contains 5 sections, 18 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Difference graphs with domination number one
  4. The diameter and dominating sets of the difference graph of a cyclic group
  5. The diameter of the difference graph of a non-cyclic nilpotent group

Key Result

Lemma 3.1

(BC22) Let $a,b\in G$ be two distinct non-trivial elements. If $a$ and $b$ commute and $\gcd(o(a),o(b))=1$, then $a\sim b$ in $\mathcal{D}(G)$.

Theorems & Definitions (20)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Corollary 3.5
  • Lemma 4.1
  • Theorem 4.2
  • Corollary 4.3
  • Corollary 4.4
  • Lemma 5.1
  • ...and 10 more