Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature

An amply regular graph is a regular graph such that any two adjacent vertices have $\alpha$ common neighbors and any two vertices with distance $2$ have $\beta$ common neighbors. We prove a sharp lower bound estimate for the Lin–Lu–Yau curvature of any amply regular graph with girth $3$ and $\beta >\alpha$. The proof involves new ideas relating discrete Ricci curvature with local matching properties: This includes a novel construction of a regular bipartite graph from the local structure and related distance estimates. As a consequence, we obtain sharp diameter and eigenvalue bounds for amply regular graphs.