Curvature and local matchings of conference graphs and extensions

We prove a conjecture of Bonini et al. on the precise values of the Lin--Lu--Yau curvature of conference graphs, i.e., strongly regular graphs with parameters $(4\gamma+1,2\gamma,\gamma-1,\gamma)$. Our method depends only on the parameter relations and applies to broader classes of amply regular graphs. In particular, we develop a new combinatorial approach to show the existence of local perfect matchings. A key observation is that counting common neighbors leads to useful quadratic polynomials. As a corollary, we derive an interesting number-theoretic result concerning quadratic residues.