A condition for non-negative Lin-Lu-Yau curvature

TL;DR

This work analyzes non-negative Lin-Lu-Yau curvature on graphs by deriving a concrete minimum-degree bound $\delta(G) \ge \frac{2|V|}{3} - 1$ that guarantees $Ric(G) \ge 0$ for finite graphs. The authors connect the Lin-Lu-Yau curvature to the α-Ollivier-Ricci curvature, using a diameter bound to reduce the problem and a transport-plan analysis with $\alpha = \frac{1}{d_x+1}$ to show $\kappa_\alpha(x,y) \ge 0$, which implies $\kappa(x,y) \ge 0$. They also demonstrate sharpness by constructing graphs with $\delta(G) = \frac{2|V|}{3} - 2$ containing edges with negative curvature $\kappa(x,y)$. The results provide a practical discrete criterion for non-negative Lin-Lu-Yau curvature and extend prior regular-graph findings to non-regular graphs, with potential implications for discrete geometric inequalities and related areas in graph theory.

Abstract

We investigate the Ollivier-Ricci curvature and its modification introduced by Lin, Lu, and Yau on locally finite graphs. The main contribution of this work is a lower bound on the minimum vertex degree of a graph ensuring non-negative Lin-Lu-Yau curvature. Additionally, we examine the sharpness of this lower bound.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Wasserstein distance
• Lemma 2.2
• Definition 2.3: Ollivier-Ricci curvature
• Definition 2.4: Lin-Lu-Yau curvature
• Theorem 2.5: Bourne2018, Theorem 4.4
• Theorem 2.6
• Remark 2.7
• Theorem 2.8: MH2024, Theorem 4.3