On the second smallest and the largest normalized Laplacian eigenvalues of a graph

TL;DR

This paper investigates how the second smallest normalized Laplacian eigenvalue and the spectral radius of the normalized Laplacian matrix change under three different graph perturbations, providing insights into spectral graph theory.

Contribution

It analyzes the effects of three specific graph operations on key spectral properties of the normalized Laplacian matrix, advancing understanding of spectral graph perturbations.

Findings

Behavior of λ₂(G) under perturbations analyzed

Spectral radius ρ(ℒ(G)) behavior studied

Results contribute to spectral graph theory understanding

Abstract

Let $G$ be a simple connected graph with order $n$. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$. Let $\lambda_{k}(G)$ be the $k$-th smallest normalized Laplacian eigenvalue of $G$. Denote $\rho(A)$ the spectral radius of the matrix $A$. In this paper, we study the behaviors of $\lambda_{2}(G)$ and $\rho(\mathcal{L}(G))$ when the graph is perturbed by three operations.