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

In 1976, Cameron, Goethals, Seidel, and Shult classified all the graphs with smallest eigenvalue at least $-2$ by relating such graphs to root systems that occur in the classification of semisimple Lie algebras. In this paper, extending their beautiful theorem, we give a complete classification of all connected graphs with smallest eigenvalue in (-λ^*, -2), where λ^* = ρ1/2 + ρ-1/2 ≈ 2.01980, and ρ is the unique real root of x³ = x + 1. Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in (-λ, -2) for any constant λ > 2.