Monotonicity of Steklov eigenvalues on graphs and applications

In this paper, we obtain monotonicity of Steklov eigenvalues on graphs which as a special case on trees extends the results of He and Hua (Steklov flows on trees and applications. arXiv:2103.07696) to higher Steklov eigenvalues and gives affirmative answers to two problems proposed in He and 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 and Hua (Calc Var Partial Differ Equ 61(3):101, 2022. arXiv:2011.11014).