Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature

TL;DR

This paper applies Alexandrov geometry to classify infinite semiplanar graphs with nonnegative curvature, establishing their volume doubling, Poincaré inequality, and polynomial growth harmonic functions, linking graph theory with geometric analysis.

Contribution

It provides a metric classification of these graphs and demonstrates key geometric analysis properties using Alexandrov geometry methods.

Findings

Graphs are classified via metric methods

Volume doubling and Poincaré inequality are established

Polynomial growth harmonic functions are characterized

Abstract

In the present paper, we apply Alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature in the sense of Higuchi. We obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point. Moreover, we show the volume doubling property and the Poincar\'e inequality on such graphs. The quadratic volume growth of these graphs implies the parabolicity. In addition, we prove the polynomial growth harmonic function theorem analogous to the case of Riemannian manifolds.