Integral Ricci Curvature for Graphs

We introduce the notion of integral Ricci curvature $I_{{\kappa }_0}$ for graphs, which measures the amount of Ricci curvature below a given threshold ${\kappa }_0$. 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 ${\lambda }_1$ of the Graph Laplacian, generalizing the results obtained by Lin, Lu, and Yau. All estimates are uniform, depending only on geometric parameters like ${\kappa }_0$, $I_{{\kappa }_0}$, $d_M$, or $D$, and do not require the graphs to be positively curved.