Bounds on curvature in regular graphs

We study the curvature-dimension inequality in regular graphs. We develop techniques for calculating the curvature of such graphs, and we give characterizations of classes of graphs with positive, zero, and negative curvature. Our main result is to compare the curvature-dimension inequality in these classes to the so-called Ollivier curvature. A consequence of our results is that in the case that the graph contains no subgraph isomorphic to either $K_3$ or $K_{2,3}$ these curvatures usually have the same sign, and we characterize the exceptions.