Higher Steklov eigenvalues of graphs on surfaces

TL;DR

This paper establishes upper bounds for higher Steklov eigenvalues of graphs on surfaces, extending spectral geometry concepts to discrete graph settings and connecting to Laplacian eigenvalues.

Contribution

It provides a novel upper bound for higher Steklov eigenvalues of graphs on surfaces using metric deformation, bridging discrete and continuous spectral geometry.

Findings

Derived upper bounds for higher Steklov eigenvalues

Connected Steklov eigenvalue bounds to Laplacian eigenvalues

Introduced a method using probability flows for spectral analysis

Abstract

In this paper, we study the higher Steklov eigenvalues of graphs on surfaces. We obtain the upper bound of higher Steklov eigenvalues of a finite graph $G$ with boundary $B$ and genus $g$ by using metrical deformation via probability flows. Our result can be regarded as a discrete analogue of Karpukhin's bound in spectral geometry. Moreover, this result implies the upper bound of higher Laplacian eigenvalues given by Kelner, Lee, Price and Teng (Geom. Funct. Anal., 2011).