  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.
