The Graph Curvature Calculator and the Curvatures of Cubic Graphs

David Cushinga, Riikka Kangaslampib* , Valtteri Lipiäinenc , Shiping Liud & George W. Staggea Department of Mathematical Sciences, Durham University, Durham, UKb Computing Sciences, Tampere University, Tampere, Finlandc Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finlandd School of Mathematical Sciences, University of Science and Technology of China, Hefei, Chinae School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, UKCONTACT Riikka Kangaslampi riikka.kangaslampi@tuni.fi Computing Sciences, Tampere University, Tampere 33014, Finland.AbstractWe 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.