Ricci-flat $5$-regular graphs

The notion of Ricci curvature of Riemannian manifolds in differential geometry has been extended to other metric spaces such as graphs. The Ollivier–Ricci curvature between two vertices of a graph can be seen as a measure of how closely connected the neighbors of the vertices are compared to the distance between them. A Ricci-flat graph is then a graph in which each edge has curvature $0$. There has been previous work in classifying Ricci-flat graphs under different definitions of Ricci curvature, notably graphs with large girth and small degrees under the definition of Lin–Lu–Yau, which is a modification of Ollivier’s definition of Ricci curvature. In this paper, we continue the effort of classifying Ricci-flat graphs and study specifically Ricci-flat $5$-regular graphs under the definition of Lin–Lu–Yau, we prove that a Ricci-flat $5$-regular symmetric graph must be isomorphic to a graph of $72$ vertices called $RF^5_{72}$.

Full Text (PDF format)

Received 7 May 2021

Accepted 29 September 2021

Published 29 March 2023