Three-point semidefinite-programming optimality for triangle-free strongly regular graphs

Prove that for every connected triangle-free strongly regular graph other than a complete bipartite graph, the spherical code obtained by embedding the graph into the eigenspace corresponding to its smallest eigenvalue is a maximal spherical code, with maximality certified by three-point semidefinite-programming bounds.

Background

The paper studies spherical codes whose inner products avoid prescribed subsets of the interval [-1,1), using linear-programming bounds to establish maximal cardinality, minimum design size, and universal energy optimality. In Section 4.5, the authors consider spherical embeddings of strongly regular graphs. Each such graph yields embeddings into eigenspaces associated with its two nontrivial eigenvalues, producing two-distance spherical codes with inner products p<0<q.

The authors note that Cohn, de Laat, and Leijenhorst used semidefinite programming to prove maximal-cardinality results for certain strongly regular graph embeddings and conjectured that a similar approach should work for many triangle-free strongly regular graphs. The paper records this as an explicit conjecture but does not resolve it; its later results instead establish universal optimality for the appropriate strongly regular graph embeddings under the condition p+q≤0.