Spherical two-distance sets and eigenvalues of signed graphs
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866909819034664960 |
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| author | Jiang, Zilin Tidor, Jonathan Yao, Yuan Zhang, Shengtong Zhao, Yufei |
| author_facet | Jiang, Zilin Tidor, Jonathan Yao, Yuan Zhang, Shengtong Zhao, Yufei |
| contents | We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $N_{α,β}(d)$ denote the maximum number of unit vectors in $\mathbb R^d$ where all pairwise inner products lie in $\{α,β\}$. For fixed $-1\leqβ<0\leqα<1$, we propose a conjecture for the limit of $N_{α,β}(d)/d$ as $d \to \infty$ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $α+2β<0$ or $(1-α)/(α-β) \in \{1, \sqrt{2}, \sqrt{3}\}$.
Our work builds on our recent resolution of the problem in the case of $α= -β$ (corresponding to equiangular lines). It is the first determination of $\lim_{d \to \infty} N_{α,β}(d)/d$ for any nontrivial fixed values of $α$ and $β$ outside of the equiangular lines setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_06633 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Spherical two-distance sets and eigenvalues of signed graphs Jiang, Zilin Tidor, Jonathan Yao, Yuan Zhang, Shengtong Zhao, Yufei Combinatorics Metric Geometry We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $N_{α,β}(d)$ denote the maximum number of unit vectors in $\mathbb R^d$ where all pairwise inner products lie in $\{α,β\}$. For fixed $-1\leqβ<0\leqα<1$, we propose a conjecture for the limit of $N_{α,β}(d)/d$ as $d \to \infty$ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $α+2β<0$ or $(1-α)/(α-β) \in \{1, \sqrt{2}, \sqrt{3}\}$. Our work builds on our recent resolution of the problem in the case of $α= -β$ (corresponding to equiangular lines). It is the first determination of $\lim_{d \to \infty} N_{α,β}(d)/d$ for any nontrivial fixed values of $α$ and $β$ outside of the equiangular lines setting. |
| title | Spherical two-distance sets and eigenvalues of signed graphs |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2006.06633 |