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A short proof of a perturbation inequality for the spectral radius

Lele Liu, Bo Ning

Abstract

Let $G$ be a simple graph, and denote by $λ(G)$ its spectral radius. Sun and Das (2020) established that for any non-isolated vertex $v$ with degree $d(v)$, \[ λ(G)\leq \sqrt{λ(G-v)^2 + 2d(v) - 1}, \] which is a conjecture original posed by Guo, Wang, and Li (2019). Sun and Das's proof uses several tools from spectral graph theory. In this short note, we provide a concise and self-contained proof of this inequality using matrix analysis.

A short proof of a perturbation inequality for the spectral radius

Abstract

Let be a simple graph, and denote by its spectral radius. Sun and Das (2020) established that for any non-isolated vertex with degree , which is a conjecture original posed by Guo, Wang, and Li (2019). Sun and Das's proof uses several tools from spectral graph theory. In this short note, we provide a concise and self-contained proof of this inequality using matrix analysis.

Paper Structure

This paper contains 2 sections, 1 theorem, 11 equations.

Table of Contents

  1. Introduction
  2. A short proof of Theorem \ref{['Thm:GuoetalConj']}

Key Result

Theorem 1

Let $G$ be a simple graph and let $v\in V(G)$ be a non-isolated vertex of degree $d(v)$. Then If $G$ is connected, the equality holds if and only if $G\cong K_{1,n-1}$ and $d(v) = 1$ or $G\cong K_n$.

Theorems & Definitions (2)

  • Theorem 1: Sun-Das2020
  • Remark 1