Spectral conditions for spherical two-distance sets
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866918321254825984 |
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| author | Noman, Iliyas Yao, Yuan |
| author_facet | Noman, Iliyas Yao, Yuan |
| contents | A set of points $S$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ is called a 2-distance set if the set of pairwise distances between the points has cardinality two. The 2-distance set is called spherical if its points lie on the unit sphere in $\mathbb{R}^{d}$. We characterize the spherical 2-distance sets using the spectrum of the adjacency matrix of an associated graph and the spectrum of the projection of the adjacency matrix onto the orthogonal complement of the all-ones vector. We also determine the lowest dimensional space in which a given spherical 2-distance set could be represented using the graph spectrum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_12582 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Spectral conditions for spherical two-distance sets Noman, Iliyas Yao, Yuan Combinatorics Metric Geometry A set of points $S$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ is called a 2-distance set if the set of pairwise distances between the points has cardinality two. The 2-distance set is called spherical if its points lie on the unit sphere in $\mathbb{R}^{d}$. We characterize the spherical 2-distance sets using the spectrum of the adjacency matrix of an associated graph and the spectrum of the projection of the adjacency matrix onto the orthogonal complement of the all-ones vector. We also determine the lowest dimensional space in which a given spherical 2-distance set could be represented using the graph spectrum. |
| title | Spectral conditions for spherical two-distance sets |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2211.12582 |