$-\lambda^* = -2.0198008871...$

$-\lambda^* = -2.0198008871...$

the limit of the smallest eigenvalue of
(refer to adjacency matrices)

Fundamental problem

Classification and characterization of graphs with bounded eigenvalues

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

$\mathcal{G}(2) = \{$ graphs with smallest eigenvalue $\ge -2 \}$

History of $\mathcal{G}(2)$

$\mathcal{G}(2) = \{$graphs with smallest eigenvalue $\ge -2 \}$

$\mathcal{G}(2) \supset \{$line graphs$\}$

$\mathcal{G}(2) \supset \{$line graphs$\}$

Hoffman · 1969 $\mathcal{G}(2) \supset \{$generalized line graphs$\}$

Hoffman · 1969 $\mathcal{G}(2) \supset \{$generalized line graphs$\}$

Observation If $G \in \mathcal{G}(2)$, then components $\in \mathcal{G}(2)$

Classify $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$

But 😕 why $\lambda^*$?

J. and Polyanskii $\mathcal{G}(\lambda) \setminus \mathcal{G}(2)$ is finite for $\lambda < \lambda^*$

*Only consider connected graphs from now on

Observation $\{$$\} \subset \mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$

$\lambda^*$ is smallest s.t. $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ is infinite 🙂

J. and Acharya
Complete classification of $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$

Low-res version

Every big graph in $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ looks like

augmented path extension (ape)

Low-res version

Every big graph in $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ looks like

augmented path extension (ape)

A notable portion of mavericks look alike

$-\lambda^{**} = -2.02124$

the limit of the smallest eigenvalue of

Low-res version

For $\lambda \in (\lambda^*, \lambda^{**})$,
eEvery big graph in $\mathcal{G}(\lambda$$^*$$) \setminus \mathcal{G}(2)$ looks like

Observation $\{$$\} \subset \mathcal{G}(\lambda^{**}) \setminus \mathcal{G}(2)$

Low-res version

For $\lambda \in (\lambda^*, \lambda^{**})$,
every big graph in $\mathcal{G}(\lambda) \setminus \mathcal{G}(2)$ looks like

Proof techniques

Low-res version

Every big graph in $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ looks like

Weaker version

Every big graph in $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ contains

Suffices to show
Every big graph in $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ contains no $F$ in

Assume that big $G$ in $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ contains $F$

Ramsey theorem for connected graphs
If big $G$ contains $F$, then $G$ contains

Observation $\not\in \mathcal{G}(\lambda^*)$

Lemma If $\not\in \mathcal{G}(\lambda^*)$, then the same holds for

Check $\not\in \mathcal{G}(\lambda^*)$ for $F$ in

Open problems

Problem A Classify graphs with smallest eigenvalues in $(-\lambda^{**},-\lambda^*)$.

Problem B Classify signed graphs with smallest eigenvalue in $(-\lambda^*,-2)$.
Signed graphs are graphs with edges labeled by $+$ or $-$
Refer to signed adjacency matrix
In particular, classify those that are big.