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. We also highlight an online tool for calculating the curvature of graphs under several variants of these curvature notions that we use in the classification. As a consequence of the classification result we show, that non-negatively curved cubic expanders do not exist.

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Cited by in corpus (6)

• Emergence of the Circle in a Statistical Model of Random Cubic Graphs
• Curvatures, graph products and Ricci flatness
• Liouville property and non-negative Ollivier curvature on graphs
• Ricci-flat cubic graphs with girth five
• Exact Expressions and Reduced Linear Programmes for the Ollivier Curvature in Graphs
• Combinatorial, Bakry-Émery, Ollivier's Ricci curvature notions and their motivation from Riemannian geometry