Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 6

Special issue in honor of Fan Chung

Guest editors: Paul Horn, Yong Lin, and Linyuan Lu

Ricci-flat $5$-regular graphs

Pages: 2511 – 2535

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n6.a8


Heidi Lei (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Shuliang Bai (Shing-Tung Yau Center, Southeast University, Nanjing, China)


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 $RF^5_{72}$.

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Received 7 May 2021

Accepted 29 September 2021

Published 29 March 2023