Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917292132007936 |
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| author | Acharya, Hricha Jiang, Zilin |
| author_facet | Acharya, Hricha Jiang, Zilin |
| contents | 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 $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$, and $ρ$ 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 $(-λ, -2)$ for any constant $λ> 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13136 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult Acharya, Hricha Jiang, Zilin Combinatorics 05C50, 05C76, 05C30 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 $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$, and $ρ$ 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 $(-λ, -2)$ for any constant $λ> 2$. |
| title | Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult |
| topic | Combinatorics 05C50, 05C76, 05C30 |
| url | https://arxiv.org/abs/2404.13136 |