Part I

Eigenvalues

Adjacency matrix and smallest eigenvalue

$\lim_n \lambda_1(P_n) = -2$

What's the limit of the smallest eigenvalue? $-\lambda^*$

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

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

Characterization of graphs with bounded eigenvalues

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

Cauchy interlacing theorem

$\mathcal{G}(\lambda)$ is closed under taking subgraphs
* all subgraphs are induced

Part II

Forbidden subgraphs

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

$\mathcal{G}(\lambda)$ is closed under taking subgraphs

Planar graphs are closed under taking minors

Wagner's theorem: no $K_5$ or $K_{3,3}$ minors

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

$\mathcal{G}(\lambda)$ is closed under taking subgraphs

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 $\ge -\lambda \}$

Question [Bussemaker and Neumaier 1992]
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

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

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

J. and Polyanskii 2022+
Yes if and only if $\lambda < \lambda^* \approx 2.01980$

Generalization to signed graphs

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

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

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

Motivation [Tidor, Yao, Zhang and Zhao 2022] Application to spherical two distance sets

Another application

For every $λ < λ^*$, there exists $n$ s.t. for every symmetric integer matrix $A$ whose diagonal entries are all zero, if $A$ has an eigenvalue less than $-λ$, then so does one of its principal submatrix of order at most $n$. The same conclusion does not hold for any $λ ≥ λ^*$

Vijayakumar: for $λ = 2$ the natural number $n$ can be as small as (but no smaller than) $10$.

Part III

Ramsey

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

$\mathcal{G}(\lambda)$ can be defined by finite forbidden subgraphs
if and only if $\lambda < \lambda^* \approx 2.01980$

Break into three cases

$\lambda < 2$
$2 \le \lambda < \lambda^*$
$\lambda \ge \lambda^*$

Case 1: For $\lambda < 2$, $\mathcal{G}(\lambda)$ can be defined by finite forbidden subgraphs

Ramsey-type result

For every $F$ and $\ell, m$ there exists $N$ s.t. for every connected graph $G$ with more than $N$ vertices, if no member in $\{K_{1,4}\} \cup \mathcal{X}(F, \ell, m)$ is a subgraph of $G$ then $F$ is not a subgraph of $G$.

Here, the extension family $\mathcal{X}(F, \ell, m)$ consists of

Step 1: Find $\ell, m$ such that both $\mathcal{X}(C, \ell, m)$ and $\mathcal{X}(D, \ell, m)$ are disjoint from $\mathcal{G}(\lambda)$

Forbid $\{K_{1,4}\} \cup \mathcal{X}(C, \ell, m) \cup \mathcal{X}(D, \ell, m)$

Fact (van Rooij and Wilf): Every graph that contains neither $C$ nor $D$ as a subgraph is a line graph

What're left to forbid?
Graphs with at most $N$ vertices and line graphs

Step 2: Forbid graphs outside $\mathcal{G}(\lambda)$ with at most $N$ vertices, $P_\ell$, some path-clique/clique extensions of cycles, and some clique extensions of cliques

What're still left to forbid?
Line graphs of rooted trees $T$, where $T$ is of depth $\le \ell$, and all vertices but root of $T$ have degree $\le m$

Step 3: Forbid $L(T)$ for $T$ that is minimal in $\mathcal{F}$, where $\mathcal{F} = \{T \colon L(T) \not\in \mathcal{G}(\lambda)$ depth of $T \le \ell$ and degrees of non-roots are $\le m\}$

Are there only finitely many minimal $T$ in $\mathcal{F}$?

Step 3.1: Enumerate bounded trees $T_1, T_2, \dots, T_n$

Step 3.2: Encode each $T \in \mathcal{F}$ by $(t_1, \dots, t_n)$ where $t_i$ is the # of occurrences of $T_i$ as a connected component in the graph obtained by removing the root from $T$.

Encodings of minimal $T$ form an antichain in $(\mathbb{N}^n, ≤)$

Step 3: Forbid $L(T)$ for $T$ that is minimal in $\mathcal{F}$, where $\mathcal{F} = \{T \colon L(T) \not\in \mathcal{G}(\lambda)$ depth of $T \le \ell$ and degrees of non-roots are $\le m\}$

Are there only finitely many minimal $T$ in $\mathcal{F}$?

Encodings of minimal $T$ form an antichain in $(\mathbb{N}^n, ≤)$

Dickson's lemma
The poset $(\mathbb{N}^n, ≤)$ does not contain infinite antichains

Case 2: For $2 \le \lambda < \lambda^*$, $G(\lambda)$ can be defined by finite forbidden subgraphs

• Similar Ramsey-type argument
• Use graphs on right in place of $C$ and $D$
• Computer assisted proof
• $G_{23}$ is critical

Open problems

Problem A: Effectivization

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

Problem C: Same problem but for signed graphs

Problem D:Forbidden principle submatrices for $\{-1,0,+1\}$-valued symmetric matrices with eigenvalues bounded from below; connection to Lehmer's Mahler measure problem