# Distance bounds for graphs with some negative Bakry-\'Emery curvature

## TL;DR

This paper establishes new distance bounds for graphs with mostly positive Bakry-Émery curvature, allowing for a finite or infinite set of vertices with non-positive curvature, thus extending previous results to non-constant curvature scenarios.

## Contribution

It introduces the first distance bounds for graphs under non-constant Bakry-Émery curvature assumptions, including cases with non-positive curvature on a finite or infinite set of vertices.

## Key findings

- Finite non-positively curved set implies explicit diameter bound
- Infinite non-positively curved set results in a subset of a tubular neighborhood
- First results assuming non-constant Bakry-Émery curvature on graphs

## Abstract

We prove distance bounds for graphs possessing positive Bakry-\'Emery 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-\'Emery curvature assumptions on graphs.