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arXiv · 2511.15205

The first Steklov eigenvalue bound for graphs of positive genus

Lixiang Chen·Yongtang Shi·Liwen Zhang
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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.

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Lixiang Chen, Yongtang Shi, Liwen Zhang. 2025-11-19. The first Steklov eigenvalue bound for graphs of positive genus. https://arxiv.org/abs/2511.15205

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