Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs

TL;DR

This paper establishes upper bounds for Steklov eigenvalues of subgraphs in polynomial growth Cayley graphs by relating them to geometric properties of associated manifolds.

Contribution

It introduces a novel method linking Cayley graph subgraphs to manifolds to derive eigenvalue bounds, advancing spectral graph theory.

Findings

Eigenvalues tend to zero proportionally to 1/|B|^{1/(d-1)}

Upper bounds are derived for the Steklov spectrum of subgraphs

Method involves discretizing manifolds and applying comparison theorems

Abstract

We study the Steklov problem on a subgraph with boundary $(\Omega,B)$ of a polynomial growth Cayley graph $\Gamma$. We prove that for each $k \in \mathbb{N}$, the $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally to $1/|B|^{\frac{1}{d-1}}$, where $d$ represents the growth rate of $\Gamma$. The method consists in associating a manifold $M$ to $\Gamma$ and a bounded domain $N \subset M$ to a subgraph $(\Omega, B)$ of $\Gamma$. We find upper bounds for the Steklov spectrum of $N$ and transfer these bounds to $(\Omega, B)$ by discretizing $N$ and using comparison Theorems.