Inner-outer curvatures, Ollivier-Ricci curvature and volume growth of graphs

We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationships of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.