Communications in Analysis and Geometry
Volume 29 (2021)
Ricci-flat cubic graphs with girth five
Pages: 1559 – 1570
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  that misses the Triplex.
Received 8 February 2018
Accepted 11 March 2019
Published 17 May 2022