The first Steklov eigenvalue bound for graphs of positive genus

TL;DR

We address the problem of bounding the first nontrivial Steklov eigenvalue $λ_2(G,δΩ)$ for finite graphs with boundary embedded on a genus $g$ surface and bounded degree. The approach adapts the circle-packing framework for higher-genus surfaces, combined with a degree $O(g)$ analytic map to the sphere to transfer energy estimates to a sphere setting, and uses hexagon-subdivision refinements together with immersion-based energy comparisons. The main result is $λ_2(G,δΩ) ≤ Oigl(g/|δΩ|igr)$, a discrete analogue of Kokarev's bound up to constants. This work strengthens links between discrete spectral graph theory, circle packings on Riemann surfaces, and topological network resistance, with potential implications for spectral bounds in topological networks.

Abstract

Let $G$ be a graph of genus $g$ with boundary $δΩ$. 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 $(G, δΩ)$, and they posed the problem of determining a corresponding bound for graphs of genus $g>0$. In this paper, we prove an $O\left(\frac{g}{|δΩ|}\right)$ 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.

Figures (3)

• Figure 1: The hexagon-subdivision refinement of two triangle.
• Figure 2: The grid graph induced by two adjacent triangles.
• Figure 3: Two initial-paths between $\pi_V(v)$ and $\pi_V(u)$.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2: Circle packing theorem
• Lemma 2.3
• Theorem 2.4
• Lemma 2.5
• Lemma 2.6
• proof : Proof of Theorem \ref{['thm:sd']}
• proof : Proof of Lemma \ref{['Lem:H']}