Integral Ricci Curvature for Graphs
Xavier Ramos Olivé
TL;DR
This work introduces integral Ricci curvature for graphs by measuring how much Lin-Lu-Yau curvature falls below a threshold $\kappa_0$ via the deficit quantities $I_{\kappa_0}$ and $I^\alpha_{\kappa_0}$. It then proves three uniform, threshold-based bounds: a Bonnet-Myers-type diameter bound, a Moore-type bound on the number of vertices in terms of the maximum degree and diameter, and a Lichnerowicz-type bound on the first nonzero eigenvalue of the normalized graph Laplacian, all expressed in terms of $\kappa_0$, $I_{\kappa_0}$, $I^\alpha_{\kappa_0}$, $d_M$, and $D$. When the integral curvature vanishes ($I_{\kappa_0}=0$, or $I^\alpha_{\kappa_0}=0$), these estimates recover Lin-Lu-Yau’s results for graphs with positive curvature, thus extending them to graphs that are not positively curved. The paper also discusses sharpness through path graphs and observes curvature-driven obstructions for hypothetical Moore graphs, highlighting the practical relevance of integral curvature in discrete geometry and data-analytic settings. Overall, the approach provides robust geometric controls under weaker curvature hypotheses, with explicit formulas that depend only on global graph parameters and the deficit terms.
Abstract
We introduce the notion of integral Ricci curvature $I_{κ_0}$ for graphs, which measures the amount of Ricci curvature below a given threshold $κ_0$. We focus our attention on the Lin-Lu-Yau Ricci curvature. As applications, we prove a Bonnet-Myers-type diameter estimate, a Moore-type estimate on the number of vertices of a graph in terms of the maximum degree $d_M$ and diameter $D$, and a Lichnerowicz-type estimate for the first eigenvalue $λ_1$ of the Graph Laplacian, generalizing the results obtained by Lin, Lu, and Yau. All estimates are uniform, depending only on geometric parameters like $κ_0$, $I_{κ_0}$, $d_M$, or $D$, and do not require the graphs to be positively curved.