Ricci-flat graphs with girth at least five

A graph is called Ricci-flat if its Ricci curvatures vanish on all edges. Here 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. Math., 2009. In this paper, we classified all Ricci-fiat connected graphs with girth at least five: they are the infinite path, cycle C-n (n >= 6), the dodecahedral graph, the Petersen graph and the half-dodecahedral graph. We also construct many Ricci-flat graphs with girth 3 or 4 by using the root systems of simple Lie algebras.