What question did this study set out to answer?

The aim is to examine Ollivier–Ricci and Lin–Lu–Yau curvatures in regular graphs and establish bounds for positivity.

February 12, 2026Open Access

Regular graphs with positive Ollivier–Ricci curvature

Key Points

The aim is to examine Ollivier–Ricci and Lin–Lu–Yau curvatures in regular graphs and establish bounds for positivity.
Derive formulas for Ollivier–Ricci and Lin–Lu–Yau curvatures based on graph parameters and optimal neighborhood assignments.
Implement efficient algorithms to compute these curvatures on regular graphs.
Analyze the relationship between the degree of regular graphs and the positivity of the curvatures.
Establish a lower bound on the degree required for positive Lin–Lu–Yau curvature and non-negative Ollivier–Ricci curvature.
Identify the maximum number of vertices possible in d-regular graphs with positive Lin–Lu–Yau curvature.

Abstract

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.

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