Ricci-flat cubic graphs with girth five

David Cushing Riikka Kangaslampi Yong Lin Shiping Liu Linyuan Lu Shing-Tung Yau

Combinatorics mathscidoc:1912.43700

arXiv preprint arXiv:1802.02982, 2018.2
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.
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@inproceedings{david2018ricci-flat,
  title={Ricci-flat cubic graphs with girth five},
  author={David Cushing, Riikka Kangaslampi, Yong Lin, Shiping Liu, Linyuan Lu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205052301491264},
  booktitle={arXiv preprint arXiv:1802.02982},
  year={2018},
}
David Cushing, Riikka Kangaslampi, Yong Lin, Shiping Liu, Linyuan Lu, and Shing-Tung Yau. Ricci-flat cubic graphs with girth five. 2018. In arXiv preprint arXiv:1802.02982. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205052301491264.
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