Classification of equiangular lines
with a fixed angle

Zilin Jiang
Arizona State University
August 7, 2026
Joint work with Theodore Gossett, Adam Teets, and Zoe Wellner

Joint work with

Theodore Gossett
Adam Teets
Zoe Wellner
  1. Equiangular lines in $\mathbb R^d$
  2. Equiangular lines with a fixed angle
  3. Equiangular lines with a fixed angle in high dimensions
  4. Classification of equiangular lines with a fixed angle

Equiangular lines in $\mathbb{R}^d$ are lines through origin
pairwise separated by the same angle

$\mathbb R^2$
$\mathbb R^3$

$N(d)$ = maximum number of equiangular lines in $\mathbb{R}^d$

$d$
22
3-43-4
55
66
77
8-138-13
14141414
1515
16161616
17171717
$N(d)$
33
66
1010
1616
2828
2828
28-3028-3028-2928-2928
3636
40-4240-4240-4140-4140
48-5048-5048-5048-4948
18181818
191919
202020
2121
2222
2323
24-4124-4124-41
424242
434343
48-6148-6148-6148-6157-6157-59
72-7672-7672-7672-7672-74
90-9690-9690-9690-9690-94
126126
176176
276276
276-861276-861276
276-903276-903276-288
276-946276-946344
  • Trivial: $d=2$
  • 1948 Haantjes for $d=3,4$
  • 1966 van Lint–Seidel for $d=5,6,7$
  • 1973 Gerzon: $N(d)\le d(d+1)/2$
    1973 Lemmens–Seidel for $d=15,21,22,23$
  • 2014 Barg–Yu for $d=24,\ldots,43$
    2016 Greaves–Koolen–Munemasa–Szöllősi for $d=14,16$
  • 2019 Greaves–Yatsyna for $d=17$
  • 2021–2023 Greaves–Syatriadi–Yatsyna for $d=14$ and $16\le d\le20$
  • 2024 Greaves–Syatriadi for $d=18$

What's the maximum number $N(d)$ of equiangular lines in $\mathbb{R}^d$?

1973 Gerzon

At most $\frac{1}{2}d(d+1)$

2000 de Caen

At least $cd^2$

Angles $\to$ 90° as $d\to\infty$

What happens if the angles are held fixed?

  1. Equiangular lines in $\mathbb R^d$
  2. Equiangular lines with a fixed angle
  3. Equiangular lines with a fixed angle in high dimensions
  4. Classification of equiangular lines with a fixed angle

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

1973 Neumann$N_{\alpha}(d) \le 2d$, unless $1/\alpha$ is odd
1973 Lemmens and
Seidel		$N_{1/3}(d) = 2,4,6,10,16$
for $d \in \{2, \dots, 6\}$, and
$\max(28, 2(d-1))$ for $d \ge 7$.
1989 Neumaier$N_{1/5}(d) = \lfloor \frac{3}{2}(d-1) \rfloor$ for $d \ge d_0$
2022 Cao, Koolen, Lin, and Yu$N_{1/5}(d) = \max(276, \lfloor 3(d-1)/2\rfloor)$ for $d \ge 23$

What's the maximum number $N_{1/5}(d)$ of
equiangular lines in $\mathbb{R}^d$ with a fixed angle $\arccos(1/5)$?

$d$2345678910111213
$N_{1/5}(d)$234679101216182026
14151617181920212223-41
2836404857-5972-7490-94126176276
$d$2345678910111213
$N(d)$366101628282828282828
14151617181920212223-41
2836404857-5972-7490-94126176276

What's the maximum number $N_{1/5}(d)$ of
equiangular lines in $\mathbb{R}^d$ with a fixed angle $\arccos(1/5)$?

$d$2345678910111213
$N_{1/5}(d)$234679101216182026
14151617181920212223-41
2836404857-5972-7490-94126176276

What's the maximum number $N_{1/5}(d)$ of
equiangular lines in $\mathbb{R}^d$ with a fixed angle $\arccos(1/5)$?

$d$2345678910111213
$N_{1/5}(d)$234679101216182026
14151617181920212223-41
2836404857-5972-7490-94126176276

Known results on $N(d)$ and $N_{1/5}(d)$ agree for $d \in \{14, \dots , 41\}$

1973 Lemmens–Seidel for $d \le 48$
$N(d) \le \max(N_{1/3}(d), N_{1/5}(d), \lfloor \frac{48d}{49-d}\rfloor)$

  1. Equiangular lines in $\mathbb R^d$
  2. Equiangular lines with a fixed angle
  3. Equiangular lines with a fixed angle in high dimensions
  4. Classification of equiangular lines with a fixed angle

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

1989 Neumaier$N_{1/5}(d) = \lfloor \frac{3}{2}(d-1) \rfloor$ for $d \ge d_0$
2016 Bukh$N_{\alpha}(d) \le c_\alpha d$
2018 Balla, Dräxler,
Keevash, Sudakov		$N_{\alpha}(d) \le 1.93d$
for $d \ge d_0(\alpha)$ if $\alpha \neq 1/3$
2020 J.–Polyanskii$N_{\alpha}(d) \le 1.49d$
for $d \ge d_0(\alpha)$ if
$\alpha \notin \{1/3, 1/(1+2\sqrt2), 1/5\}$

2016 Bukh's conjecture

$N_{1/(2k-1)}(d) \approx \frac{kd}{k-1}$

2020 J.–Polyanskii conjecture

$N_{\alpha}(d) \approx \frac{kd}{k-1}$, where $k = k(\lambda)$, $\lambda = \frac{1-\alpha}{2\alpha}$.

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

$\alpha$$\lambda$$G$$k$$N_\alpha(d)$
$\tfrac{1}{3}$$1$$2$$2d$
$\tfrac{1}{1+2\sqrt2}$$\sqrt 2$$3$$\tfrac{3d}{2}$
$\tfrac{1}{5}$$2$$3$$\tfrac{3d}{2}$
$\frac{1}{7}$$3$$4$$\tfrac{4d}{3}$
me
Jonathan Tidor
Yufei Zhao
Yuan Yao
Shengtong Zhang

2021 J., Tidor, Yao, Zhang, Zhao

$N_\alpha(d) = \lfloor \frac{k}{k-1}(d-1) \rfloor$ for $d \ge d_0(\alpha)$ if $k(\lambda) < \infty$;
$N_\alpha(d) = d+o(d)$ otherwise.

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

Equiangular lines
Graph
unit vectors
vertices
$\langle v_i,v_j\rangle=-\alpha$
$v_i\sim v_j$
$\langle v_i,v_j\rangle=\alpha$
$v_i\nsim v_j$
Gram matrix $\succeq0$
$M=\lambda I-A_G+\frac12J\succeq0$
rank is at most $d$
$\operatorname{rank}M\le d$
$v_i\leftrightarrow -v_i$
switching equivalence

$M=\lambda I-A_G+\frac12J\succeq0$
Given $n=|G|$, minimize $\operatorname{rank}M$

Generic

$G=G_1\sqcup\cdots\sqcup G_t$

$\rho(G_i)\le\lambda$

Take $G_i$ to be a smallest graph with $\rho(G_i)=\lambda$

$\operatorname{rank}M\approx n-\frac nk$

Exceptional

$G$ connected

bounded degree

$\lambda_2(G)=\lambda$

$\operatorname{rank}M\approx n-\operatorname{mult}(\lambda,G)$

2021 J., Tidor, Yao, Zhang, Zhao

For every bounded-degree, connected $n$-vertex graph $G$,
$\operatorname{mult}(\lambda_2(G),G)=o(n)$.

Exceptional graphs are eventually outperformed

  1. Equiangular lines in $\mathbb R^d$
  2. Equiangular lines with a fixed angle
  3. Equiangular lines with a fixed angle in high dimensions
  4. Classification of equiangular lines with a fixed angle

New phenomena

Generic constructions do not just win eventually

For certain angles,
exceptional constructions are bounded in order

Motivation

$\alpha$$\lambda$$G$$k(\lambda)$$N_\alpha(d)$Status
$\tfrac{1}{3}$$1$$2$$\max(28,2(d-1))$
for $d\ge7$		known for all $d$
$\tfrac{1}{1+2\sqrt2}$$\sqrt2$$3$$\left\lfloor 3(d-1)/2\right\rfloor$
for $d\ge d_0$		?
$\tfrac{1}{5}$$2$$3$$\max\left(276,\left\lfloor 3(d-1)/2\right\rfloor\right)$
for $d\ge23$		known for all $d$ but
18, 19, 20

Classification at $\alpha^*=1/(1+2\sqrt2)$

2026 Gossett, J., Teets, Wellner

$d$234567891011121314
$N_{\alpha^*}(d)$234681014151617182022
$N_{\alpha^*}(d)=\max\left(24,\left\lfloor 3(d-1)/2\right\rfloor\right)$ for $d\ge15$

Classification: up to switching, there are exactly 63,673 exceptional constructions, each with at most 28 vertices

Generic constructions

$P_3$$\langle P_3\rangle$$\langle P_3\rangle_\pm$
$+$$+$

take induced subgraphs and unions

take switching equivalence

Five forbidden seeds

Every graph outside $\langle P_3\rangle_\pm$ contains
up to switching, one of these five induced subgraphs

A minimal forbidden graph has at most 8 vertices

  1. Start from the five forbidden seeds
  2. Add one vertex with every possible neighborhood
  3. Keep $\lambda I-A_G+\frac12J\succeq0$
  4. Deduplicate up to switching

What we needed versus what we found

$\alpha=1/(1+2\sqrt2),\qquad \lambda=\sqrt2$

Needed for asymptotics

Bound second-eigenvalue multiplicity

$\Downarrow$

Exceptional constructions are eventually outperformed

What classification revealed

No exceptional graph has order $>28$

$\Downarrow$

Exceptional constructions are absolutely bounded

The stronger phenomenon was surprising

Open problem

Let $\mathcal G(\lambda)$ be the connected graphs with spectral radius at most $\lambda$

For which $\lambda$ are there only finitely many graphs outside
$\langle\mathcal G(\lambda)\rangle_\pm$ with $\lambda I-A_G+J/2\succeq0$?

  • Yes for $\lambda=\sqrt2$: 63,673 exceptions, each with at most 28 vertices
  • Yes for $\lambda=1$: 902 exceptions, each with at most 28 vertices
  • No for certain $\lambda$ with $k(\lambda)=\infty$
    2023 Schildkraut: $\lambda=2\sqrt{u^2+1}-1$ for any integer $u\ge10^5$

One last theorem

Theorem

This was a great workshop.

Proof. Left as an exercise to the participants.

Many thanks to the organizers for making it happen.

Zilin Jiang
Arizona State University
[email protected]