We determine the maximum number $N_α(d)$ of equiangular lines with fixed angle $\arccosα$ for $α= 1/(1+2\sqrt2)$ in $d$-dimensional Euclidean space: $2,3,4,6,8,10,14,15,16,17,18,20,22$ for $d \in \{2,\dots,14\}$, and $\max(24, \lfloor 3(d-1)/2 \rfloor)$ for $d \ge 15$. This appears to be the first complete determination of $N_α(d)$ in all dimensions $d$ for a fixed nontrivial $α$, since the work of Lemmens and Seidel for $α= 1/3$ in 1973.
We derive a procedure for computing an upper bound on the number of equiangular lines in various Euclidean vector spaces by generalizing the classical pillar decomposition developed by (Lemmens and Seidel, 1973); namely, we use linear algebra and combinatorial arguments to bound the number of vectors within an equiangular set which have inner products of certain signs with a negative clique. After projection and rescaling, such sets are also certain spherical two-distance sets, and semidefinite programming techniques may be used to bound the size. Applying our method, we prove new relative bou…
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 $0 < α< 1$. Let $N_α(d)$ denote the maximum number of lines through the origin in $\mathbb{R}^d$ with pairwise common angle $\arccos α$. Let $k$ denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly $(1-α)/(2α)$. If $k < \infty$, then $N_α(d) = \lfloor k(d-1)/(k-1) \rfloor$ for all sufficiently large $d$, and otherwise $N_α(d) = d…
School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle $θ$ and sufficiently large $n$ there are at most $2n-2$ lines in $\mathbb{R}^n$ with common angle $θ$. Moreover, this is achievable only for $θ= \arccos(1/3)$. We also show that for any set of $k$ fixed angles, one can find at most $O(n^k…
In this paper we show that the conjecture of Lemmens and Seidel of 1973 for systems of equiangular lines with common angle $\arccos (1/5)$ is true. Our main tool is forbidden subgraphs for smallest Seidel eigenvalue $-5$.
The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let $\mathcal{F}(λ)$ be the family of connected graphs of spectral radius $\le λ$. We show that $\mathcal{F}(λ)$ can be defined by a finite set of forbidden subgraphs if and only if $λ< λ^* := \sqrt{2+\sqrt{5}} \approx 2.058$ and $λ\not\in \{α_2, α_3, \dots\}$, where $α_m = β_m^{1/2} + β_m^{-1/2}$ and $β_m$ is the largest root of $x^{m+1}=1+x+\dots+x^{m-1}$. The study of forbidden subgraphs characterization for $\mathcal{F}(λ)$ is motivated by the problem of estimating the maximum cardinality of equiangular lines…
New bounds for equiangular lines and Balla's conjecture
School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
Answering a question of Jiang and Polyanskii as well as Jiang, Tidor, Yao, Zhang, and Zhao, we show the existence of infinitely many angles $θ$ for which the maximum number of lines in $\mathbb R^n$ meeting at the origin with pairwise angles $θ$ exceeds $n+Ω(\log\log n)$ but is at most $n+o(n)$. To accomplish this, we construct, for various real $λ$ and integer $d$, $d$-regular graphs with second eigenvalue exactly $λ$ and arbitrarily large second eigenvalue multiplicity. Central to our construction is a distribution on factors of bipartite graphs which possesses concentration properties.
A set of lines through the origin in Euclidean space is called equiangular when any pair of lines from the set intersects with each other at a common angle. We study the maximum size of equiangular lines in Euclidean space and use graph theoretic approach to prove that all the currently known construction for maximum equiangular lines in $\mathbb R^d$ cannot add another line to form a larger equiangular set of lines if $14 \leq d \leq 20$ and $d \neq 15$. We give new constructions of large equiangular lines which are 248 equiangular lines in $\mathbb R^{42}$, 200 equiangular lines in $\mathbb{…
An $L$-spherical code is a set of Euclidean unit vectors whose pairwise inner products belong to the set $L$. We show, for a fixed $α,β>0$, that the size of any $[-1,-β]\cup\{α\}$-spherical code is at most linear in the dimension. In particular, this bound applies to sets of lines such that every two are at a fixed angle to each another.
The absolute upper bound on the number of equiangular lines that can be found in $\mathbf{R}^d$ is $d(d+1)/2$. Examples of sets of lines that saturate this bound are only known to exist in dimensions $d=2,3,7$ or $23$. By considering the additional property of incoherence, we prove that there exists a set of equiangular lines that saturates the absolute bound and the incoherence bound if and only if $d=2,3,7$ or $23$. This allows us classify all tight spherical $5$-designs $X$ in $\mathbf{S}^{d-1}$, the unit sphere, with the property that there exists a set of $d$ points in $X$ whose pairwise …
A set of lines through the origin is called equiangular if every pair of lines defines the same angle, and the maximum size of an equiangular set of lines in $\mathbb{R}^n$ was studied extensively for the last 70 years. In this paper, we study analogous questions for $k$-dimensional subspaces. We discuss natural ways of defining the angle between $k$-dimensional subspaces and correspondingly study the maximum size of an equiangular set of $k$-dimensional subspaces in $\mathbb{R}^n$. Our bounds extend and improve a result of Blokhuis.
In this paper, claims by Lemmens and Seidel in 1973 about equiangular sets of lines with angle $1/5$ are proved by carefully analyzing pillar decompositions, with the aid of the uniqueness of two-graphs on $276$ vertices. The Neumann Theorem is generalized in the sense that if there are more than $2r-2$ equiangular lines in $\mathbb{R}^r$, then the angle is quite restricted. Together with techniques on finding saturated equiangular sets, we determine the maximum size of equiangular sets "exactly" in an $r$-dimensional Euclidean space for $r = 8$, $9$, and $10$.
A set of lines in $\mathbb{R}^d$ passing through the origin is called equiangular if any two lines in the set form the same angle. We proved an alternative version of the three-point semidefinite constraints developed by Bachoc and Vallentin, and the multi-point semidefinite constraints developed by Musin for spherical codes. The alternative semidefinite constraints are simpler when the concerned object is a spherical $s$-distance set. Using the alternative four-point semidefinite constraints, we found the four-point semidefinite bound for equiangular lines. This result improves the upper boun…
In this paper we explicitly estimate the number of points in a subset $A \subset \R^{d}$ as a function of the maximum angle $\angle A$ that any three of these points form, provided $\angle A < θ_d := \arccos(-\frac 1 {d}) \in (π/2,π)$. We also show $\angle A < θ_d$ ensures that $A$ coincides with the vertex set of a convex polytope. This study is motivated by a question of Paul Erdős and indirectly by a conjecture of László Fejes Tóth.
This thesis is a study of large sets of unit vectors in $\cx^n$ such that the absolute value of their standard inner products takes on only a small number of values. We begin with bounds: what is the maximal size of a set of lines with only a given set of angles? We rederive a series of upper bounds originally due to Delsarte, Goethals and Seidel, but in a novel way using only zonal polynomials and linear algebra. In the process we get some new results about complex $t$-designs and also some new characterizations of tightness. Next we consider constructions. We describe some generic constructi…
