Bakry-Emery curvature and diameter bounds on graphs

TL;DR

This paper establishes diameter bounds for graphs with positive Bakry-Emery Ricci curvature, including a sharp bound for hypercubes and a novel Bonnet-Myers type theorem for unbounded graph Laplacians, improving previous results.

Contribution

It introduces new diameter bounds based on Bakry-Emery curvature, notably a Bonnet-Myers type theorem for unbounded graph Laplacians, expanding geometric analysis on graphs.

Findings

Sharp diameter bound for hypercubes

First Bonnet-Myers type theorem for unbounded graph Laplacians

Improved diameter bounds over previous results

Abstract

We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hypercubes. The other result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet-Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from [6] and [10].