# Hardy-Muckenhoupt Bounds for Laplacian Eigenvalues

## TL;DR

This paper introduces new graph-based bounds for Laplacian eigenvalues using Hardy-type inequalities, providing constant factor estimates for Dirichlet and Neumann eigenvalues.

## Contribution

The paper develops novel graph quantities and techniques to estimate Laplacian eigenvalues, advancing spectral graph theory methods.

## Key findings

- Psi(G,S) and Psi_2(G) effectively estimate eigenvalues
- Hardy-type inequalities are applied in a discrete setting
- Provides constant factor bounds for eigenvalues

## Abstract

We present two graph quantities Psi(G,S) and Psi_2(G) which give constant factor estimates to the Dirichlet and Neumann eigenvalues, lambda(G,S) and lambda_2(G), respectively. Our techniques make use of a discrete Hardy-type inequality.