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 the 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.arXiv:1712.03033v2 [math.CO]

Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature

“…Note furthermore that Chang stands for any one of the three Chang graphs and that the Doob graphs are given by Doob n,m = K n 4 ×Shk m with n, m ≥ 1, where Shk denotes the Shrikhande graph, and the (7, 2)-Kneser, Conway-Smith graph and Hall graph are the three locally Petersen graphs. The curvatures inf x∼y κ(x, y) were determined with the help of the curvature calculator [7] at http://www.mas.ncl.ac.uk/graph-curvature/ Proposition 6.5. A distance-regular graph with second largest adjacency eigenvalue θ 1 = b 1 − 1 and µ = 1 cannot be Lichnerowicz sharp.…”

Curvatures, graph products and Ricci flatness

“…The curvatures of these examples were calculated numerically via the interactive web‐application at https://www.mas.ncl.ac.uk/graph-curvature/. For more details about this very useful tool we refer the readers to [9].…”

Long-Scale Ollivier Ricci Curvature of Graphs

“…Beyond recent theoretical work on this notion (see [1,2,3,4,10,16]), there have been several applications outside mathematics such as in biology (see [5,18,23]) and in computing (see [13,21,22] ).…”