Bounding the Diameter and Eigenvalues of Amply Regular Graphs via Lin–Lu–Yau Curvature

Abstract

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.