Higher Steklov eigenvalues of graphs on surfaces
Xiongfeng Zhan, Zhe You
TL;DR
This work extends the Steklov spectral theory to graphs embedded on surfaces by establishing upper bounds for higher Steklov eigenvalues $\sigma_k$ using a discrete analogue of metric deformation via probability flows. The authors develop a framework combining Rayleigh quotients, probability flows, and padded decompositions to relate spreading weights and flow congestion through convex duality under Slater's condition. Their main results show genus-dependent bounds $\sigma_k \\lesssim D g( ext{log} g)^2 rac{k^2}{|B|}$, with planar cases simplifying to $\\mathcal{O}(D rac{k^2}{|B|})$ and minor-free analogues, aligning discrete estimates with continuous Karpukhin-type bounds. This yields a concrete, scalable approach to bounding higher Steklov eigenvalues on graphs on surfaces and highlights the role of surface topology (genus) and graph decomposability (padded partitions) in spectral behavior.
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. This result can be regarded as a discrete analogue of Karpukhin's bound in spectral geometry.