Three-point bounds for spectral embeddings of triangle-free strongly regular graphs

Establish that for every connected, triangle-free strongly regular graph G other than the complete bipartite graph Kn,n, the spectral embedding C of G into the eigenspace of its adjacency matrix corresponding to the smallest eigenvalue is an optimal spherical code, and demonstrate that the Bachoc–Vallentin three-point semidefinite programming bounds certify this optimality by achieving equality.

Background

Triangle-free strongly regular graphs yield two-distance spherical codes via spectral embedding into the smallest-eigenvalue eigenspace of the adjacency matrix. The paper proves optimality for several such codes (Hoffman–Singleton, Gewirtz, and M22) using three-point bounds and suggests that this approach should apply to all connected triangle-free strongly regular graphs beyond Kn,n.

The conjecture aims to unify these case-by-case proofs into a general theorem asserting that three-point semidefinite programming bounds are sharp for the entire family, thereby certifying optimality of the corresponding spherical codes.