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

In 1976, Cameron, Goethals, Seidel, and Shult classified all the graphs whose smallest eigenvalue is at least $-2$ by relating such graphs to root systems that appear in the classification of semisimple Lie algebras. In this paper, extending their beautiful theorem, we give a complete classification of all connected graphs whose smallest eigenvalue lies in (\! -λ ^*, -2) , where λ ^* = ho 1/2 + ho -1/2 ≈ 2.01980 , and $ho$ 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 .