Existence and uniqueness of discrete Einstein metrics on trees is established under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory, with a sharp upper bound on the largest eigenvalue and the implication that positive-curvature cases require caterpillar trees.
Ricci-flat cubic graphs with girth five.Comm
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Discrete Einstein metrics on trees
Existence and uniqueness of discrete Einstein metrics on trees is established under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory, with a sharp upper bound on the largest eigenvalue and the implication that positive-curvature cases require caterpillar trees.