Curvature and local matchings of conference graphs and extensions

TL;DR

This paper proves a conjecture about the Lin–Lu–Yau curvature of conference graphs, introduces a new combinatorial approach for local perfect matchings, and connects graph properties to number theory.

Contribution

It provides a proof of a specific curvature conjecture for conference graphs and develops a novel combinatorial method applicable to broader amply regular graphs.

Findings

Confirmed the curvature values for conference graphs.

Established a new combinatorial approach for local perfect matchings.

Derived a number-theoretic result related to quadratic residues.

Abstract

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.