Regular graphs with positive Ollivier–Ricci curvature

Abstract We study the Ollivier–Ricci curvature and its modification introduced by Lin, Lu, and Yau on regular graphs. We derive precise formulas for the Ollivier–Ricci curvature and the Lin–Lu–Yau curvature on regular graphs, expressed in terms of graph parameters and an optimal assignment between neighborhoods. Using these precise formulas, we provide an efficient implementation for computing the Ollivier–Ricci curvature and the Lin–Lu–Yau curvature on regular graphs, enabling us to explore the distribution of regular graphs with positive Lin–Lu–Yau curvature. Our main contribution is establishing a lower bound on the degree of a regular graph that guarantees positive Lin–Lu–Yau curvature and non-negative Ollivier–Ricci curvature, along with an analysis of the sharpness of this bound. Additionally, we study the maximal number of vertices that an d -regular graph with positive Lin–Lu–Yau curvature can have.