The Graph Curvature Calculator and the Curvatures of Cubic Graphs

Abstract

We classify all cubic graphs with either non-negative Ollivier-Ricci curvature or non-negative Bakry-Émery curvature everywhere. We show in both curvature notions that the non-negatively curved graphs are the prism graphs and the Möbius ladders. As a consequence of the classification result we show that non-negatively curved cubic expanders do not exist. We also introduce the Graph Curvature Calculator, an online tool developed for calculating the curvature of graphs under several variants of the curvature notions that we use in the classification.

Keywords:

• discrete curvature
• graph theory
• cubic graph
• expander graph
• Graph Curvature Calculator

AMS SUBJECT CLASSIFICATION:

• Primary 05C50
• Secondary 39A12, 51K10

Additional information

Funding

The authors are grateful to Prof. Norbert Peyerimhoff for his continued support and guidance. DC and SL would like to acknowledge that this work was supported by the EPSRC Grant EP/K016687/1 “Topology, Geometry and Laplacians of Simplicial Complexes.” DC would like to thank Aalto University and RK and SL Durham University for their hospitality during research visits, where some of the above work was carried out. DC wants to thank the EPSRC for financial support through his postdoctoral prize.