The first Steklov eigenvalue bound for graphs of positive genus

TL;DR

This paper establishes an upper bound for the first Steklov eigenvalue in graphs of positive genus, extending previous results from genus zero to higher genus graphs with bounded degree.

Contribution

It provides the first bound for graphs of positive genus, generalizing prior genus-zero results and aligning with a continuous analogue by Kokarev.

Findings

Bound is proportional to g/|δΩ| for graphs of genus g

Extends Steklov eigenvalue bounds to higher genus graphs

Matches continuous bounds up to a constant factor

Abstract

Let $G$ be a graph of genus $g$ with boundary $\delta\Omega$. For $g=0$, Lin and Zhao [J. Lond. Math. Soc. 112 (2025), Paper No. e70238] proved an upper bound for the first (non-trivial) Steklov eigenvalue of $(G, \delta\Omega )$, and they posed the problem of determining a corresponding bound for graphs of genus $g>0$. In this paper, we prove an $O\left(\frac{g}{|\delta \Omega|}\right)$ bound for a bounded-degree graph of positive genus $g$. Our result can be regarded as a discrete analogue of Kokarev's bound [Adv. Math. 258 (2014), 191-239], up to a constant factor.