What's the maximum number of equiangular lines in $\mathbb{R}^n$?

Angles $\to$ 90° as $n\to\infty$

What happens if the angles are held fixed?

What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?

What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?

Spectral radius order $k(\lambda) := $ smallest $k$ such that
$\exists$ $k$-vertex graph $G$ whose adjacency matrix has spectral radius $\lambda$

What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?

me

Jonathan Tidor

Yufei Zhao

Yuan Yao

Shengtong Zhang

Spectral radius order $k(\lambda) := $ smallest $k$ such that
$\exists$ $k$-vertex graph $G$ whose adjacency matrix has spectral radius $\lambda$

1. When $k < \infty$, proof needs $n \ge 2^{2^{C\lambda k}}$.
   Balla showed $n \ge 2^{k^{Ck}}$; conjecturally $n \ge k^C$.
2. When $k = \infty$, $o(n) = O_{\alpha}(n/\log\log n)$.
   Schildkraut constructed $\Omega_{\alpha}(\log\log n)$.

Connection to Spectral Graph Theory

Equiangular lines in $\mathbb{R}^n$ with angle $\arccos\alpha$
$\Leftrightarrow$ Unit vectors in $\mathbb{R}^n$ with inner product $\pm \alpha$
$\Leftarrow$$\Rightarrow$ Graph $G$, where $u \sim v$ in $G$ iff $\langle u, v\rangle = -\alpha$
s.t. $(1-\alpha) I - 2\alpha A_G + \alpha J \succeq 0$ and rank $\le n$

Equivalently, given $\alpha$ and $n$, find $G$ with max order s.t.
$(1-\alpha) I - 2\alpha A_G + \alpha J \succeq 0$ and rank $\le n$

Example when $\alpha = 1/5$, take $\lfloor (n-1)/2 \rfloor$ copies of

Upper bound on $E_\alpha(n)$

Given $\alpha$ and $n$, find $G$ with max order s.t.
$M := (1-\alpha) I - 2\alpha A_G + \alpha J \succeq 0$ and rank $\le n$

Rank-nullity
$E_\alpha(n) \le n + \mathrm{null}(M)$$\le \mathrm{null}((1-\alpha) I - 2\alpha A_G) + n + 1$

Since $M \succeq 0$, $\lambda := \frac{1-\alpha}{2\alpha}$ could be

1. largest eigenvalue of $G$ (equality case)
2. second largest eigenvalue of connected $G$ (need to rule out)

$\lambda := \frac{1-\alpha}{2\alpha}$ could be second largest eigenvalue of connected $G$

Switching $v$ to $-v$ in $\mathbb{R}^n$

Sublinear Second Eigenvalue Multiplicity

Proof Sketch

Multiplicity of second largest eigenvalue $\lambda$ of an $n$-vertex connected graph $G$ of bounded degree is at most $Cn/\log\log n$.

$r = c\log\log n$ and $s = c \log n$

• $H$ is $G$ with small $r$-net removed
• Radius $s$ balls in $H$ have spectral radius $\le (\lambda^{2r} - 1)^{1/2r}$
  Vertex removal lowers spectral radius
• Bound $\mathrm{mult}(\lambda, H)$ via moments
• $\mathrm{mult}(\lambda, G) \le \mathrm{mult}(\lambda, G) + \lvert r\text{-net} \rvert = o(n)$
  by Cauchy interlacing

What is the maximum second eigenvalue multiplicity of
an $n$-vertex connected graph of bounded degree?

McKenzie, Rasmussen, Srivastava 2021
For regular graphs, $O(n/\log^c n)$

What is the maximum second eigenvalue multiplicity of
an $n$-vertex Cayley graph of bounded degree?

Gromov, Colding–Minicozzi, Kleiner
Lee–Makarychev: Constant for abelian groups

What is the maximum number of unit vectors in $\mathbb{R}^n$ such that pairwise inner products are either $\alpha$ or $\beta$?

Spherical Two-distance Sets

$E_{\alpha, \beta}(n) = $ maximum number of unit vectors in $\mathbb{R}^n$
such that pairwise inner products are $\beta < 0 < \alpha$

Bukh 2005: $E_{\alpha, \beta}(n) = O_{\alpha,\beta}(n)$

Problem Determine $\lim_{n\to\infty} E_{\alpha, \beta}(n) /n$

Conjecture Depend on $k_p(\lambda)$ in terms of signed graphs
where $p = \lfloor -\alpha/\beta\rfloor+1$ and $\lambda = (1-\alpha)/(\alpha - \beta)$

Solved in special cases

• J., Tidor, Yao, Zhang, Zhao 2023: $p \le 2$ or $\lambda = 1,\sqrt{2},\sqrt{3}$
• J., Polyanskii 2025+: $\lambda < \lambda^* \approx 2.0198$
  $\lambda^* := \sqrt{\rho} + 1/\sqrt{\rho}$, where $\rho$ is root of $x^3 = x + 1$

Solution Framework

1. Local forbidden subgraphs: leverage Gram matrix $\succeq 0$
2. Global graph structure: Ramsey theory
   • Equiangular lines: bounded degree graph
   • Spherical two-distance sets:
     complete $p$-partite XOR bounded degree

Need new insights in spectral graph theory:

• Sublinear eigenvalue multiplicity is false for signed graphs
• J., Polyanskii 2025+: local forbidden subgraphs alone can never break $\lambda^*$ barrier
• J., Wang 2025+: local forbidden subgraphs + global graph structure can never break $\lambda^*$ barrier as soon as $p \ge 3$

Classification of graphs
with eigenvalues $\ge -\lambda$

• Cameron, Goethals, Seidel, and Shult 1976: $\lambda = 2$
• J., Acharya 2025: $\lambda = \lambda^* \approx 2.0198$
• Problem Signed graphs with eigenvalues $\ge -\lambda^*$

Symmetric integer matrix

• Vijayakumar 1987: Every symmetric hollow integer matrix has eigenvalue $<-2$ must have principal submatrix of order $\le 10$ with same property
• J. 2024: Determined $\lambda$ such that every every symmetric hollow integer matrix has eigenvalue $<-\lambda$ must have principal submatrix of bounded order with same property
  Problem Symmetric integer matrix with spectral radius $\le \lambda$
• Lehmer's conjecture, McKee and Smyth 2012
  Problem Classify all indecomposable integer symmetric matrices whose spectral radius in $(2, \sqrt{2 + \sqrt{5}})$

Other "Equiangular Lines"

Fixed angle: determine $\lim_{n\to\infty} E_\alpha^\mathbb{C}(n)/n$

Equiangular subspaces: configurations of $k$-dimensional subspaces in $\mathbb{R}^d$ with given pairwise angles