  We study a modified notion of Ollivier's coarse Ricci curvature on graphs
introduced by Lin, Lu, and Yau in [11]. 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 one. 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 girth 4 and 5 using elementary methods. 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.
