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Equiangular lines with a fixed angle
- Publication Year :
- 2019
- Publisher :
- arXiv, 2019.
-
Abstract
- Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix $0 < \alpha < 1$. Let $N_\alpha(d)$ denote the maximum number of lines through the origin in $\mathbb{R}^d$ with pairwise common angle $\arccos \alpha$. Let $k$ denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly $(1-\alpha)/(2\alpha)$. If $k < \infty$, then $N_\alpha(d) = \lfloor k(d-1)/(k-1) \rfloor$ for all sufficiently large $d$, and otherwise $N_\alpha(d) = d + o(d)$. In particular, $N_{1/(2k-1)}(d) = \lfloor k(d-1)/(k-1) \rfloor$ for every integer $k\ge 2$ and all sufficiently large $d$. A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.<br />Comment: 11 pages. Fixed a minor issue at the end of the proof of Theorem 1.2
- Subjects :
- Degree (graph theory)
Sublinear function
Spectral graph theory
Spectral radius
010102 general mathematics
Metric Geometry (math.MG)
0102 computer and information sciences
01 natural sciences
Combinatorics
Mathematics (miscellaneous)
Integer
Mathematics - Metric Geometry
010201 computation theory & mathematics
Bounded function
FOS: Mathematics
Mathematics - Combinatorics
Adjacency matrix
Combinatorics (math.CO)
0101 mathematics
Statistics, Probability and Uncertainty
Equiangular lines
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....55d5d2a70c5143f524f5d426e480fd6d
- Full Text :
- https://doi.org/10.48550/arxiv.1907.12466