Communications in Analysis and Geometry

Volume 29 (2021)

Number 7

Ricci-flat cubic graphs with girth five

Pages: 1559 – 1570



David Cushing (Department of Mathematical Sciences, Durham University, Durham, United Kingdom; and School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, United Kingdom)

Riikka Kangaslampi (Computing sciences, Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland)

Yong Lin (Yau Mathematical Sciences Center and Department of Mathematics, Tsinghua University, Beijing, China)

Shiping Liu (School of Mathematical Sciences, University of Science and Technology of China, Hefei, China)

Linyuan Lu (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)

Shing-Tung Yau (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


We classify all connected, simple, $3$-regular graphs with girth at least $5$ that are Ricci-flat. We use the definition of Ricci curvature on graphs given in Lin–Lu–Yau, Tohoku Math. J., 2011, which is a variation of Ollivier, J. Funct. Anal., 2009. A graph is Ricci-flat, if it has vanishing Ricci curvature on all edges. We show, that the only Ricci-flat cubic graphs with girth at least $5$ are the Petersen graph, the Triplex and the dodecahedral graph. This will correct the classification in [8] that misses the Triplex.

Received 8 February 2018

Accepted 11 March 2019

Published 17 May 2022