Equiangular lines via nodal domains
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915386252853248 |
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| author | Ge, Chuanyuan Liu, Shiping |
| author_facet | Ge, Chuanyuan Liu, Shiping |
| contents | For given $Δ>0$ and $0<λ<3/\sqrt{2}$, we show that the maximum multiplicity that $λ$ can appear as the second largest eigenvalue of a connected graph with maximum degree at most $Δ$ is $O_{Δ,λ}(1)$. This result answers a question due to Jiang, Tidor, Yao, Zhang and Zhao [Question 6.4, Ann. of Math. (2) 194 (2021), no. 3, 729-743] in the case of $0<λ<3/\sqrt{2}$, and consequently leads to improvements in their results on equiangular lines. Our proof is based on the concept of nodal domains of eigenfunctions. Indeed, we establish a multiplicity estimate in terms of maximum degree and cyclomatic number of the graph, via a novel construction of eigenfunctions with large number of nodal domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equiangular lines via nodal domains Ge, Chuanyuan Liu, Shiping Combinatorics Differential Geometry Spectral Theory For given $Δ>0$ and $0<λ<3/\sqrt{2}$, we show that the maximum multiplicity that $λ$ can appear as the second largest eigenvalue of a connected graph with maximum degree at most $Δ$ is $O_{Δ,λ}(1)$. This result answers a question due to Jiang, Tidor, Yao, Zhang and Zhao [Question 6.4, Ann. of Math. (2) 194 (2021), no. 3, 729-743] in the case of $0<λ<3/\sqrt{2}$, and consequently leads to improvements in their results on equiangular lines. Our proof is based on the concept of nodal domains of eigenfunctions. Indeed, we establish a multiplicity estimate in terms of maximum degree and cyclomatic number of the graph, via a novel construction of eigenfunctions with large number of nodal domains. |
| title | Equiangular lines via nodal domains |
| topic | Combinatorics Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2507.09511 |