### Forbidden subgraphs and spherical two-distance sets

Arizona State University
June 21, 2022
Joint work with Alexandr Polyanskii, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao

### $\lambda^* = \sqrt{\beta}+1/\sqrt{\beta} =$ 2.019800887...

where $\beta$ is real root of $x^3 = x + 1$.

Forbidden subgraphs

Spherical two-distance sets

#### Adjacency matrix, and smallest eigenvalue

 $G$ $A_G$ $\lambda_1(G)$ $K_n$ $J - I$ $-1$

Classification of graphs with bounded eigenvalues

$\mathcal{G}(\lambda) = \{$ graphs with smallest eigenvalue $\ge -\lambda \}$

Observation: Cauchy interlacing theorem implies
$\mathcal{G}(\lambda)$ is closed under taking subgraphs
* all subgraphs are induced

Question: Define $\mathcal{G}(\lambda)$ by forbidden subgraphs?

$\mathcal{G}(\lambda) = \{$ graphs with smallest eigenvalue $\ge -\lambda \}$

Question: Define $\mathcal{G}(\lambda)$ by forbidden subgraphs?

Simple case $\lambda < 1$

$\lambda_1(K_2) = -1$ hence $K_2 \not\in \mathcal{G}(\lambda)$

$\mathcal{G}(\lambda) = \{$ graphs with no edges $\}$

Tautology: Forbid all graphs outside $\mathcal{G}(\lambda)$

$\mathcal{G}(\lambda) = \{$ graphs with smallest eigenvalue $\ge -\lambda \}$

Bussemaker and Neumaier 1992: For which $\lambda \in \mathbb{R}$ can $\mathcal{G}(\lambda)$ be defined by finite forbidden subgraphs?

$\mathcal{G}(2)$ is complex; it contains all the line graphs

Cameron, Goethals, Seidel, and Shult: For every connected graph, it is in $\mathcal{G}(2)$ if and only if it is represented by a subset of $D_n$ or $E_8$ (root systems).

Kumar, Rao and Singhi $\mathcal{G}(2)$ can be defined by forbidden subgraphs with $\le 10$ vertices

Bussemaker and Neumaier 1992: For which $\lambda \in \mathbb{R}$ can $\mathcal{G}(\lambda)$ be defined by finite forbidden subgraphs?

"however, these seem to be very difficult problems"

J. and Polyanskii 2022+: For $\lambda < \lambda^* \approx 2.01980$, yes; otherwise, no.

### Generalization to signed graphs

$\mathcal{G}^\pm(\lambda) = \{$ signed graphs with smallest e.v. $\ge -\lambda\}$

Question: Define $\mathcal{G}^\pm(\lambda)$ by forbidden subgraphs?

J. and Polyanskii 2022+ Yes if and only if $\lambda < \lambda^*$

$G^\mp(\lambda) = \{$ signed graphs with largest e.v. $\le \lambda\}$

Question: Define $\mathcal{G}^\mp(\lambda)$ by forbidden subgraphs?

Corollary Yes if and only if $\lambda < \lambda^*$

Application For every $λ < λ^*$, there exists $n$ s.t.
for every symmetric integer matrix$$A = \begin{bmatrix}0 & * & \dots & * \\ * & 0 & \dots & * \\ \vdots & \vdots & \ddots & \vdots \\ * & * & \cdots & 0 \end{bmatrix},$$if $\lambda_1(A_0) \ge -λ$ for every principal submatrix $A_0$ of order at $\le n$, then $\lambda_1(A) \ge -λ$.
Vijayakumar 1987: for $λ = 2$, $n = 10$

Forbidden subgraphs

Spherical two-distance sets

### Spherical two-distance sets

$\{ v_1, \dots, v_N \in \mathbb{R}^d \colon$ each $v_i$ is a unit vector
and $\langle v_i, v_j \rangle = \alpha$ or $\beta\}$

Fix $-1 \le \beta < 0 \le \alpha < 1$

$N_{\alpha, \beta}(d) =$ max size of spherical $\{\alpha,\beta\}$-code in $\mathbb{R}^d$

$N_{\alpha, \beta}(d) =$ max size of spherical $\{\alpha,\beta\}$-code in $\mathbb{R}^d$

Problem: Determine $N_{\alpha, \beta}(d)$ for large $d$
In particular, determine $\lim_{d \to \infty} N_{\alpha, \beta}(d)/d$

J., Tidor, Yao, Zhang, and Zhao 2021
For "equiangular" case $\beta = -\alpha$ $$N_{\alpha, -\alpha} = \frac{k}{k-1}d + O(1)$$ where $k$ is "spectral radius order" of $\frac{1-\alpha}{2\alpha}$

$N_{\alpha, \beta}(d) =$ max size of spherical $\{\alpha,\beta\}$-code in $\mathbb{R}^d$

Problem: Determine $\lim_{d \to \infty} N_{\alpha, \beta}(d)/d$

J., Tidor, Yao, Zhang, Zhao 2022

• Lower bound on $N_{\alpha, \beta}(d)$, which depends only on $p := \lfloor -\alpha/\beta \rfloor + 1$ and $\lambda = \frac{1-\alpha}{\alpha-\beta}$
• Matching upper bound for large $d$ when $p \in \{1,2\}$ or $\lambda \in \{1, \sqrt2, \sqrt3\}$; e.g. $N_{2/5,-1/5} = 3d + O(1)$
• Conjectured lower bound is tight for large $d$

How to get an upper bound on $N_{\alpha, \beta}(d)$?

Step 1: Associate a graph $G$ to a spherical $\{\alpha,\beta\}$-code

Step 2: There exists $\Delta$ s.t. $G$ after removing at most $\Delta$ vertices, is a $\Delta$-modification of a complete $p$-partite graph $K$

A graph $G$ is a $∆$-modification of another graph $H$ on the same vertex set if the symmetric difference of $G$ and $H$ has maximum degree at most $∆$

Step 3: Work with signed graph $G^\pm$ with signed adjacency matrix $A_G - A_K$

Step 1: Associate a graph $G$ to a spherical $\{\alpha,\beta\}$-code

Step 2: There exists $\Delta$ s.t. $G$ after removing at most $\Delta$ vertices, is a $\Delta$-modification of a complete $p$-partite graph $K$

Step 3: Work with signed graph $G^\pm$ with signed adjacency matrix $A_G - A_K$

Forbidden subgraph framework
Given a finite family of $\mathcal{H}$ of signed graphs with largest eigenvalue $>\lambda$, can choose $\Delta$ s.t. $G^\pm$ does not contain any member in $\mathcal{H}$ as a subgraph

$\mathcal{G}^\mp(\lambda) = \{$ signed graphs with largest e.v. $\le \lambda\}$

$\mathcal{G}^\mp(\lambda)$ can be defined by finite forbidden signed subgraphs $\mathcal{H}$ if and only if $\lambda < \lambda^*$

Step 1: Associate a graph $G$ to a spherical $\{\alpha,\beta\}$-code

Step 2: There exists $\Delta$ s.t. $G$ after removing at most $\Delta$ vertices, is a $\Delta$-modification of a complete $p$-partite graph $K$

Step 3: Work with signed graph $G^\pm$ with signed adjacency matrix $A_G - A_K$

Forbidden subgraph framework
Given a finite family of $\mathcal{H}$ of signed graphs with largest eigenvalue $>\lambda$, can choose $\Delta$ s.t. $G^\pm$ does not contain any member in $\mathcal{H}$ as a subgraph

For $\lambda < \lambda^*$, can choose $\mathcal{H}$ and $\Delta$ s.t.
largest eigenvalue of $G^\pm$ is $\le \lambda$

J. and Polyanskii 2022+: Matching upper bound for large $d$ when $\lambda < \lambda^*$

### Open problems

Problem A: Classify all the connected graphs with smallest eigenvalue in $(-λ^*,-2)$. In particular, classify those that have sufficiently many vertices.

Problem B: Same problem but for signed graphs

Problem C: Prove matching upper bound for $N_{\alpha, \beta}(d)$ when $p \ge 3$ and $\lambda \ge \lambda^*$

Arizona State University
zilinj@asu.edu