The Lemmens-Seidel conjecture and forbidden subgraphs

Equiangular lines and the Lemmens-Seidel conjecture

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 $10$.

基大小为5的Lemmens-Seidel猜想

The Lemmens-Seidel conjecture for base size $5$

本文用支柱法证明了基大小为5的Lemmens-Seidel猜想，并指出Greaves等人发现的57条等角线是Lin-Yu证明的反例，同时证明这些集合是强极大的，否定了Cao等人的唯一性猜想。

Lemmens-Seidel猜想的一个新证明

A new proof of the Lemmens-Seidel conjecture

针对具有公共夹角$\u000barccos(1/5)$的等角线最大数量的Lemmens-Seidel猜想，本文提出了一个基于图特征值重数界的新证明，还为公共夹角为$\u000barccos(1/3)$的经典结果给出了新证。

Equiangular Lines and Spherical Codes in Euclidean Space

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…

具有固定角度arccos(1/(1+2√2))的等角线分类

Classification of equiangular lines with fixed angle $\arccos(1/(1+2\sqrt2))$

本文确定了在固定角度arccos(1/(1+2√2))下，不同维度d中等角线的最大数量，并给出了具体数值和公式。

New Upper Bounds for Equiangular Lines by Pillar Decomposition

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…