Ricci-flat cubic graphs with girth five
Combinatorics
2023-10-26 v1
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., 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 Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the Triplex.
Keywords
Cite
@article{arxiv.1802.02982,
title = {Ricci-flat cubic graphs with girth five},
author = {David Cushing and Riikka Kangaslampi and Yong Lin and Shiping Liu and Linyuan Lu and Shing-Tung Yau},
journal= {arXiv preprint arXiv:1802.02982},
year = {2023}
}