Geometric and spectral properties of locally tessellating planar graphs

TL;DR

This paper investigates the geometric and spectral characteristics of locally tessellating planar graphs, establishing bounds on their Cheeger constant and growth rates based on combinatorial curvature, with implications for Laplacian spectra.

Contribution

It introduces new bounds relating geometric properties like curvature to spectral features of planar graphs, enhancing understanding of their global structure.

Findings

Bounds for Cheeger constant derived from curvature

Exponential growth estimates based on local geometry

Spectral implications for graph Laplacians discussed

Abstract

In this article, we derive bounds for values of the global geometry of locally tessellating planar graphs, namely, the Cheeger constant and exponential growth, in terms of combinatorial curvatures. We also discuss spectral implications for the Laplacians.