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Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult
- Part of
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- Journal:
- Combinatorics, Probability and Computing , First View
- Published online by Cambridge University Press:
- 28 November 2025, pp. 1-23
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- Article
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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 
$\lambda \gt 2$.
