Subcubic graphs
maximum degree at most 3

Eigenvalues of adjacency matrix
$\lambda_1 \ge \dots \ge \lambda_n$

Median eigenvalues
$\lambda_h = \lambda_{\lfloor (n+1)/2 \rfloor}, \lambda_l = \lambda_{\lceil (n+1)/2 \rceil}$

Hückel Model Theory

Fowler & Pisanski 2010

Computations Most chemical graphs have
median eigenvalues in $[-1,1]$, with single exception

Eigenvalues $-3, (-\sqrt2)^6, (\sqrt2)^6, 3$

Conjecture Median eigenvalues of all but finite chemical graphs are in $[-1,1]$

Optimality
Guo & Mohar constructed infinitely many bipartite chemical graphs with median eigenvalues $\pm 1$

Fowler & Pisanski 2010 Subcubic trees

Mohar 2013 Planar bipartite chemical graphs

Mohar 2016 Bipartite chemical graphs except Heawood

Several other supplementary results

Acharya, Jeter, J., 2025 All chemical graphs except Heawood

• 1% of proof for 99% of cases
• 99% of proof for 1% of cases

For simplicity, only focus on $\lambda_h \le 1$

Proof for 99%

Take maximum cut $(A, B)$ of $G$

Additional assumption $\lvert A\rvert < \lvert B\rvert$

Observation Maximum degree of $G[B] \le 1$,
and $\lambda_1(G[B]) \le 1$

Cauchy interlacing $\lambda_{1 + \lvert A \rvert}(G) \le 1$, and $\lambda_h(G) \le 1$.

Proof for 1%

Maximum cut $(A, B)$ satisfies $\lvert A\rvert = \lvert B\rvert$

Idea 1 Move $k$ vertices $C \subset A$ s.t.
$\lambda_k(G[B \cup C]) \le 1$

Cauchy $\lambda_{k + \lvert A \setminus C \rvert}(G) \le 1$, and $\lambda_h(G) \le 1$

Note Most components of $G[B \cup C]$ are independent edges

Goal 1 Finish with tail reducer $C$

Idea 1 Move $k$ vertices $C \subset A$ s.t.
$\lambda_k(G[B \cup C]) \le 1$

Goal 1 Finish with tail reducer $C$

Idea 2 Cut-set of $(A \oplus D, A \oplus D)$ cannot be bigger than that of $(A, B)$

Goal 2 Contradiction through cut enhancer $D$

Goal 1 Finish with tail reducer $C$

Goal 2 Contradiction through cut enhancer $D$

Lemma If $ab$ is an edge of $G[A]$, then degree of $a$ is $3$, or $\{ a \}$ is tail reducer

54 pages, 170 figures

All arguments are "local"

Positive fraction $\varepsilon n$ middle eigenvalues $\subset [-1, 1]$

Further problems

Mohar & Tayfeh-Rezaie 2015 Median eigenvalues of every connected bipartite $G$ with maximum degree at most $d$ are in $[-\sqrt{d-2}, \sqrt{d-2}]$, unless $G$ is incidence graph of projective plane of order $d-1$

Optimality of $\sqrt{d-2}$ for $d \ge 4$?

Remove bipartiteness? Mohar $\sqrt{d}$.

Improve to $\sqrt{d-1}$? Improve to $\sqrt{d-2}$?