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Elementary proofs of the diameter bounds for the power graphs

Marco Barbieri, Kamilla Rekvényi

Abstract

We give a simplified version of the proofs that, outside of their isolated vertices, the complement of the enhanced power graph and of the power graph are connected of diameter at most $3$.

Elementary proofs of the diameter bounds for the power graphs

Abstract

We give a simplified version of the proofs that, outside of their isolated vertices, the complement of the enhanced power graph and of the power graph are connected of diameter at most .

Paper Structure

This paper contains 3 sections, 2 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Proof of \ref{['thm:main']}
  3. Proof of \ref{['corollary']}

Key Result

Theorem 1

Let $G$ be a finite group. Then, outside of its isolated vertices, the complement of the enhanced power graph of $G$ is connected, and its diameter is at most $3$.

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 2
  • proof : Proof of \ref{['thm:main']}
  • Remark 3: CGS, Theorem 2.12
  • proof : Proof of \ref{['corollary']}