We study a modified notion of Ollivier’s coarse Ricci curvature on graphs introduced
by Lin, Lu, and Yau. We establish a rigidity theorem for complete graphs that shows
a connected finite simple graph is complete if and only if the Ricci curvature is
strictly greater than 1. We then derive explicit Ricci curvature formulas for strongly
regular graphs in terms of the graph parameters and the size of a maximal matching
in the core neighborhood. As a consequence we are able to derive exact Ricci
curvature formulas for strongly regular graphs of girths 4 and 5 using elementary
means. An example is provided that shows there is no exact formula for the Ricci
curvature for strongly regular graphs of girth 3 that is purely in terms of graph
parameters.
