On the $l^p$ spectrum of Laplacians on graphs

TL;DR

This paper investigates the conditions under which the spectra of Laplacians on graphs are independent of the parameter p, providing criteria based on growth conditions and measure assumptions, with applications to normalized Laplacians and planar tessellations.

Contribution

It introduces a new sufficient criterion for p-independence of Laplacian spectra based on uniform subexponential growth, and explores spectral properties under various geometric and measure conditions.

Findings

Spectral p-independence holds under uniform subexponential growth.

One-sided spectral inclusion is established under mild measure assumptions.

Connections between spectral properties and geometric features like curvature are demonstrated.

Abstract

We study the $p$-independence of spectra of Laplace operators on graphs arising from regular Dirichlet forms on discrete spaces. Here, a sufficient criterion is given solely by a uniform subexponential growth condition. Moreover, under a mild assumption on the measure we show a one-sided spectral inclusion without any further assumptions. We study applications to normalized Laplacians including symmetries of the spectrum and a characterization for positivity of the Cheeger constant. Furthermore, we consider Laplacians on planar tessellations for which we relate the spectral $p$-independence to assumptions on the curvature.