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arXiv · 1802.02982

Ricci-flat cubic graphs with girth five

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

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David Cushing, Riikka Kangaslampi, Yong Lin, Shiping Liu, Linyuan Lu, Shing-Tung Yau. 2018-02-08. Ricci-flat cubic graphs with girth five. https://doi.org/10.4310/cag.2021.v29.n7.a3

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