A Comparative Study of Curvature on Trees
Sawyer Jack Robertson
TL;DR
This paper compares different notions of discrete curvature on graphs, providing formulas, comparison results, and specific theorems for trees, advancing understanding of graph geometric properties.
Contribution
It offers explicit formulas for three types of graph curvature, comparison results, and new theorems related to trees, enhancing the theoretical framework of discrete curvature.
Findings
Formulas for three types of graph curvature
Comparison results between different curvature notions
Degree-diameter theorem for trees
Abstract
There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute 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.
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Taxonomy
TopicsDigital Image Processing Techniques · Computational Geometry and Mesh Generation · Data Management and Algorithms