Characterizing the second smallest eigenvalue of the normalized Laplacian of a tree

TL;DR

This paper investigates the properties of the second smallest eigenvalue of the normalized Laplacian in trees, introducing a new concept called Perron component and establishing a monotonicity theorem for harmonic eigenfunctions.

Contribution

It introduces the Perron component for the normalized Laplacian and characterizes the second smallest eigenvalue using this concept, along with a monotonicity theorem for harmonic eigenfunctions.

Findings

Monotonicity theorem for harmonic eigenfunctions of  of the normalized Laplacian

Introduction of Perron component for the normalized Laplacian

Characterization of the second smallest eigenvalue using Perron components

Abstract

In this paper we show a monotonicity theorem for the harmonic eigenfunction of \lambda_{1} of the normalized Laplacian over the points of articulation of a graph. We introduce the definition of Perron component for the normalized Laplacian matrix of a graph and show how its second smallest eigenvalue can be characterized using this definition.