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

Abstract 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 $(\! -\lambda ^*, -2)$ , where $\lambda ^* = ho ^{1/2} + ho ^{-1/2} \approx 2.01980$ , and $ho$ is the unique real root of $x^3 = x + 1$ . Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in $(\! -\lambda , -2)$ for any constant $\lambda \gt 2$ .