Monotonicity of Steklov eigenvalues on graphs and applications

TL;DR

This paper proves the monotonicity of Steklov eigenvalues on graphs, especially trees, extending previous results to higher eigenvalues and providing new estimates relevant for graph analysis.

Contribution

It extends the monotonicity results of Steklov eigenvalues to higher eigenvalues on graphs and trees, addressing open problems and generalizing existing estimates.

Findings

Monotonicity of higher Steklov eigenvalues on graphs and trees.

Generalized isodiametric estimates for Steklov eigenvalues.

Affirmative solutions to open problems in Steklov eigenvalue theory.

Abstract

In this paper, we obtain monotonicity of Steklov eigenvalues on graphs which as a special case on trees extends the results of He-Hua [Calc. Var. Partial Differential Equations 61 (2022), no. 3, Paper No. 101, arXiv: 2103.07696] to higher Steklov eigenvalues and gives affirmative answers to two problems proposed in He-Hua [arXiv: 2103.07696]. As applications of the monotonicity of Steklov eigenvalues, we obtain some estimates for Steklov eigenvalues on trees generalizing the isodiametric estimate for the first positive Steklov eigenvalues on trees in He-Hua [arXiv:2011.11014].