论文标题
签名图的球形两距离集和特征值
Spherical two-distance sets and eigenvalues of signed graphs
论文作者
论文摘要
我们研究了确定具有两个固定角度（一个急性和一个钝角）的球形两距离集的最大尺寸的问题。令$ n_ {α，β}（d）$表示$ \ mathbb r^d $中最大的单位向量数，其中所有成对的内部产品都位于$ \ {α，β\} $中。对于固定的$ -1 \leqβ<0 \leqα<1 $，我们提出了$ n_ {α，β}（d）/d $ as $ d \ to $ d \ to \ infty $的猜想，就签名图的特征值而言。当$α+2β<0 $或$（1-α）/（α-β）\ in \ {1，\ sqrt {2}，\ sqrt {3} \} $时，我们确定此限制。 在$α=-β$的情况下，我们的工作是基于我们最近解决该问题的解决方案（对应于等法线）。这是$ \ lim_ {d \ to \ infty} n_ {α，β}（d）/d $的首次确定，对于$α$和$α$和$β$的任何非平地固定值以外的均值固定值。
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.
