Abstract

Abstract We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-Émery curvature assumptions on graphs.

Highlights

  • We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive

  • In Riemannian geometry, diameter bounds for complete connected Riemannian manifolds are well established under several curvature assumptions

  • There, the authors assumed that the amount of the Ricci curvature of the manifold M below a positive level is locally uniformly Lp-small for some p > dim M/, and obtain a diameter bound depending on this kind of smallness of the curvature

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Summary

Introduction

In Riemannian geometry, diameter bounds for complete connected Riemannian manifolds are well established under several curvature assumptions. In a highly celebrated paper, Erbar, Kuwada and Sturm proved that on metric measure spaces, the concepts of Ricci curvature via Γ-calculus (Bakry-Émery) and optimal transport and entropy (Lott-Sturm-Villani) coincide [10]. There have been attempts to generalize the Bonnet-Myers theorem to variable Ricci curvature bounds in an integral sense, see [25] and the references therein. For connected graphs G = (V , E), the authors of [28] show a sharp diameter bound assuming positive Bakry-Émery curvature in the CD(K, N)-setting for N ∈ We have and the graph admits an upper bound Degmax for the weighted vertex diamd(G) ≤

Degmax K
Setting and main result
CD conditions and semigroups
Hence we have
We estimate
Let us denote
Kt dt

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