Curvature calculations for antitrees

TL;DR

This paper demonstrates that certain infinite antitree graphs exhibit positive curvature in multiple discrete curvature frameworks, but lack global positive curvature bounds, highlighting nuanced geometric properties.

Contribution

It proves that antitrees with specific growth conditions have positive curvature in various discrete settings, expanding understanding of curvature in infinite graphs.

Findings

Antitrees exhibit positive curvature in Bakry-Émery and Ollivier-Ricci frameworks.

These graphs lack global positive curvature bounds, contrasting with classical geometric expectations.

Different techniques are required to prove results in each curvature setting.

Abstract

In this article we prove that antitrees with suitable growth properties are examples of infinite graphs exhibiting strictly positive curvature in various contexts: in the normalized and non-normalized Bakry-\'Emery setting as well in the Ollivier-Ricci curvature case. We also show that these graphs do not have global positive lower curvature bounds, which one would expect in view of discrete analogues of the Bonnet-Myers theorem. The proofs in the different settings require different techniques.