Cheeger type inequalities associated with isocapacitary constants on graphs

TL;DR

This paper develops Cheeger type inequalities using isocapacitary constants to estimate various eigenvalues and spectral bounds on finite and infinite subgraphs of graphs, extending classical spectral estimates.

Contribution

It introduces new Cheeger type inequalities based on isocapacitary constants for estimating eigenvalues and spectral bounds on graphs, including higher-order Steklov eigenvalues.

Findings

Estimates for first Dirichlet, Neumann, and Steklov eigenvalues.

Bounds for the bottom of the spectrum of Laplace and Dirichlet-to-Neumann operators.

Results for higher-order Steklov eigenvalues on finite and infinite subgraphs.

Abstract

In this paper, we introduce Cheeger type constants via isocapacitary constants introduced by Maz'ya to estimate first Dirichlet, Neumann and Steklov eigenvalues on a finite subgraph of a graph. Moreover, we estimate the bottom of the spectrum of the Laplace operator and the Dirichlet-to-Neumann operator for an infinite subgraph. Estimates for higher-order Steklov eigenvalues on a finite or infinite subgraph are also proved.