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 $(4γ+1,2γ,γ-1,γ)$. 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.

💡 Research Summary

The paper addresses a long‑standing conjecture of Bonini et al. concerning the exact Lin‑Lu‑Yau (LLY) curvature of conference graphs, which are strongly regular graphs with parameters ((4\gamma+1,,2\gamma,,\gamma-1,,\gamma)). The authors develop a purely combinatorial method that relies only on the parameter relations of amply regular graphs, and they use this method to prove the conjecture for all conference graphs, including the small cases (\gamma\le6) that require separate verification.

The starting point is the known upper bound for LLY curvature on a (d)-regular graph: \