On Discrete Curvatures of Trees

Abstract: There are several interweaving notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. Since each of these ingredients is well understood when the underlying graph is a tree, in this note, we review existing and obtain new formulas for three different types of curvature on graphs. Along the way we obtain: a comparison result for the curvatures under consideration, a degree-diameter theorem for trees, and a combinatorial identity for certain sums of distances on trees.