### Forbidden subgraphs and spherical two-distance sets

Arizona State University
October 23, 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 "generalized line graph" or "exceptional graph".

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: Can $\mathcal{G}^\pm(\lambda)$ be defined by finite 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: Can $\mathcal{G}^\mp(\lambda)$ be defined by finite forbidden subgraphs?

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

Application For every $λ < λ^*$, there exists $n_0$ s.t.
for every symmetric integer matrix$$A = \begin{bmatrix}0 & * & \dots & * \\ * & 0 & \dots & * \\ \vdots & \vdots & \ddots & \vdots \\ * & * & \cdots & 0 \end{bmatrix},$$$\lambda_1(A) \ge -λ$$\Leftarrow$$\Rightarrow$$\lambda_1(A_0) \ge -λ for every principal submatrix A_0 of order at most n_0 Vijayakumar 1987: for λ = 2, n_0 = 10 Forbidden subgraphs Spherical two-distance sets ### Spherical two-distance sets \{ v_1, \dots, v_N \in \mathbb{R}^d \colon each v_i is a unit vector and \langle v_i, v_j \rangle = \alpha or \beta\} Fix -1 \le \beta < 0 \le \alpha < 1 N_{\alpha, \beta}(d) = max size of spherical \{\alpha,\beta\}-code in \mathbb{R}^d N_{\alpha, \beta}(d) = max size of spherical \{\alpha,\beta\}-code in \mathbb{R}^d Problem: Determine N_{\alpha, \beta}(d) for large d In particular, determine \lim_{d \to \infty} N_{\alpha, \beta}(d)/d J., Tidor, Yao, Zhang, and Zhao 2021 For "equiangular" case \beta = -\alpha$$N_{\alpha, -\alpha}(d) = \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}(d) = 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^*$. Breaking the$\lambda^*\$ barrier.

Arizona State University
zilinj@asu.edu