A new proof of the Lemmens-Seidel conjecture
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914138808123392 |
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| author | Ge, Chuanyuan Liu, Shiping |
| author_facet | Ge, Chuanyuan Liu, Shiping |
| contents | In this paper, we give a new proof of the Lemmens-Seidel conjecture on the maximum number of equiangular lines with a common angle $\arccos(1/5)$. This conjecture was previously resolved by Cao, Koolen, Lin, and Yu in 2022 through an analysis involving forbidden subgraphs for the smallest Seidel eigenvalue $-5$. Our new proof is based on bounds on eigenvalue multiplicities of graphs with degree no larger than $14$. To control the maximum degree of the graph associated with equiangular lines, we employ a recent inequality of Balla derived by matrix projection techniques. Our strategy also leads to a new proof for the classical result obtained by Lemmens and Seidel in 1973 for the case where the common angle is $\arccos(1/3)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new proof of the Lemmens-Seidel conjecture Ge, Chuanyuan Liu, Shiping Combinatorics Metric Geometry Spectral Theory In this paper, we give a new proof of the Lemmens-Seidel conjecture on the maximum number of equiangular lines with a common angle $\arccos(1/5)$. This conjecture was previously resolved by Cao, Koolen, Lin, and Yu in 2022 through an analysis involving forbidden subgraphs for the smallest Seidel eigenvalue $-5$. Our new proof is based on bounds on eigenvalue multiplicities of graphs with degree no larger than $14$. To control the maximum degree of the graph associated with equiangular lines, we employ a recent inequality of Balla derived by matrix projection techniques. Our strategy also leads to a new proof for the classical result obtained by Lemmens and Seidel in 1973 for the case where the common angle is $\arccos(1/3)$. |
| title | A new proof of the Lemmens-Seidel conjecture |
| topic | Combinatorics Metric Geometry Spectral Theory |
| url | https://arxiv.org/abs/2511.03396 |