Zilin Jiang
Arizona State University
October 27, 2024
Joint work with Hricha Acharya
Hricha Acharya
the limit of the smallest eigenvalue of
(refer to adjacency matrices)
Classification and characterization of graphs with bounded eigenvalues
$\mathcal{G}(\lambda) = \{$ graphs with smallest eigenvalue $\ge -\lambda \}$
$\mathcal{G}(2) = \{$ graphs with smallest eigenvalue $\ge -2 \}$
$\mathcal{G}(2) = \{$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 $\in \mathcal{G}(2)$
Cameron, Goethals, Seidel, and Shult · 1976
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 $\{$$\} \subset \mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$
$\lambda^*$ is smallest s.t. $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ is infinite 🙂
J. and Acharya
Complete classification of $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$
Low-res version
Every big graph in $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ looks like
augmented path extension (ape)
Low-res version
Every big graph in $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ looks like
augmented path extension (ape)
Questions | Answers |
Classify $F_R$? | 794, at most 14 vertices |
Describe $\ell$? | For $F_R$, $\ell \ge \ell_0$ |
Enumerate non-apes? | 4752 mavericks at most 19 vertices |
A notable portion of mavericks look alike
the limit of the smallest eigenvalue of
Low-res version
For $\lambda \in (\lambda^*, \lambda^{**})$,
eEvery big graph in $\mathcal{G}(\lambda$$^*$$) \setminus \mathcal{G}(2)$ looks like
Observation $\{$$\} \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
Question | Answer |
Classify $F_R$? | Infinite |
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
Weaker version
Every big graph in $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$ contains
Cvetković, Doob, and Simić · 1981
Rao, Singhi, and Vijayan · 1981
If $G$ is not a generalized line graph, then $G$ 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
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.