Curvature, geometry and spectral properties of planar graphs

TL;DR

This paper introduces a curvature function for planar graphs and explores how curvature influences their geometric and spectral properties, extending tessellation results to general planar graphs.

Contribution

It defines a curvature measure for planar graphs and links curvature conditions to geometric structure, spectral bounds, and eigenfunction properties.

Findings

Non-positive curvature implies the graph is infinite and tessellation-like.

Negative curvature bounds the growth of distance balls and Cheeger's constant.

Non-positive curvature leads to absence of finitely supported eigenfunctions.

Abstract

We introduce a curvature function for planar graphs to study the connection between the curvature and the geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite and locally similar to a tessellation. We use this to extend several results known for tessellations to general planar graphs. For non-positive curvature, we show that the graph admits no cut locus and we give a description of the boundary structure of distance balls. For negative curvature, we prove that the interiors of minimal bigons are empty and derive explicit bounds for the growth of distance balls and Cheeger's constant. The latter are used to obtain lower bounds for the bottom of the spectrum of the discrete Laplace operator. Moreover, we give a characterization for triviality of essential spectrum by uniform decrease of the curvature. Finally, we show that non-positive curvature implies absence of finitely supported eigenfunctions for nearest neighbor operators.