A short proof of a perturbation inequality for the spectral radius

TL;DR

This paper provides a concise, self-contained proof of a spectral radius perturbation inequality for graphs, originally conjectured by Guo, Wang, and Li, using matrix analysis techniques.

Contribution

It offers a new, streamlined proof of a spectral radius inequality in graph theory, simplifying previous complex methods.

Findings

Established the inequality rlier conjectured by Guo, Wang, and Li.

Provided a concise proof using matrix analysis.

Confirmed the inequality holds for all simple graphs with non-isolated vertices.

Abstract

Let $G$ be a simple graph, and denote by $\lambda(G)$ its spectral radius. Sun and Das (2020) established that for any non-isolated vertex $v$ with degree $d(v)$, \[ \lambda(G)\leq \sqrt{\lambda(G-v)^2 + 2d(v) - 1}, \] 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.