Distance Bounds for Graphs with Some Negative Bakry-Émery Curvature

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.

Keywords

• bakry-émery curvature
• discrete bonnet-myers theorem
• intrinsic metric
• heat semigroup
• primary: 51k10
• secondary: 51f99, 05c12