Comparisons of Dirichlet, Neumann and Laplacian eigenvalues on graphs and their applications

TL;DR

This paper compares Dirichlet, Neumann, and Laplacian eigenvalues on graphs, explores their properties, and applies these comparisons to derive estimates and extend existing results in spectral graph theory.

Contribution

It provides new comparisons and inequalities for eigenvalues on graphs, extending classical results and connecting different types of eigenvalues in a unified framework.

Findings

Derived inequalities for Dirichlet and Neumann eigenvalues

Extended spectral estimates to Steklov eigenvalues

Connected eigenvalue comparisons to existing literature

Abstract

In this paper, we obtain some comparisons of the Dirichlet, Neumann and Laplacian eigenvalues on graphs. We also discuss their rigidities and some of their applications including some Lichnerowicz-type, Fiedler-type and Friedman-type estimates for Dirichlet eigenvalues and Neumann eigenvalues. The comparisons on Neumann eigenvalues can be translated to comparisons on Steklov eigenvalues in our setting. So, some of the results can be viewed as extensions for parts of the works of \cite{HHW} by Hua-Huang-Wang, and parts of our previous works \cite{SY,SY2}.