Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult

The classification of graphs with smallest eigenvalues at least −2 culminated in a beautiful theorem of Cameron, Goethals, Seidel and Shult, who related such graphs to root systems from the representation theory of semisimple Lie algebras. In this talk, I will explore graphs with smallest eigenvalues between −2 and −λ*, where λ* is about 2.0198, and I will explain why the mysterious number λ* is a barrier for classification. Joint work with Alexander Polyanskii and Hricha Acharya.