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# Communications in Analysis and Geometry

## Volume 29 (2021)

### Number 7

### Ricci-flat cubic graphs with girth five

Pages: 1559 – 1570

DOI: https://dx.doi.org/10.4310/CAG.2021.v29.n7.a3

#### Authors

#### Abstract

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