Upper bounds of Steklov eigenvalues on graphs

TL;DR

This paper establishes new upper bounds for the first Steklov eigenvalue on graphs, relating it to graph genus, crossing number, and boundary vertex degrees, advancing understanding of spectral properties in discrete structures.

Contribution

It introduces novel upper bounds for Steklov eigenvalues on graphs based on geometric and combinatorial parameters, extending prior continuous and discrete spectral bounds.

Findings

Bound $\sigma_2 = ext{O}(rac{ riangle(g+1)^3}{|B|})$ for graphs of genus $g$ with boundary size $|B|$.

Proves $\sigma_2 ext{max} rac{8 riangle+4X}{|B|}$ based on planar crossing number $X$.

Shows $\sigma_2 ext{max} rac{|B|}{|B|-1} imes ext{min degree of boundary vertices}$.

Abstract

Let $\Delta$ and $B$ be the maximum vertex degree and a subset of vertices in a graph $G$ respectively. In this paper, we study the first (non-trivial) Steklov eigenvalue $\sigma_2$ of $G$ with boundary $B$. Using metrical deformation via flows, we first show that $\sigma_2 = \mathcal{O}\left(\frac{\Delta(g+1)^3}{|B|}\right)$ for graphs of orientable genus $g$ if $|B| \geq \max\{3 \sqrt{g},|V|^{\frac{1}{4} + \epsilon}, 9\}$ for some $\epsilon > 0$. This can be seen as a discrete analogue of Karpukhin's bound. Secondly, we prove that $\sigma_2 \leq \frac{8\Delta+4X}{|B|}$ based on planar crossing number $X$. Thirdly, we show that $\sigma_2 \leq \frac{|B|}{|B|-1} \cdot \delta_B$, where $\delta_B$ denotes the minimum degree for boundary vertices in $B$. At last, we compare several upper bounds on Laplacian eigenvalues and Steklov eigenvalues.