OP-00321 Solution claimed

A universal central limit law for quantum-graph nodal surplus

Let $\Gamma$ be a finite connected compact metric graph whose positive edge lengths are linearly independent over $\mathbb{Q}$, with the standard (Neumann–Kirchhoff) Laplacian: $-u''$ on each edge, continuity at the vertices, and zero sum of outgoing derivatives at each vertex. Number its eigenvalues $0 = \lambda_1 \le \lambda_2 \le \cdots$. Call an index $n$ generic if $\lambda_n$ is simple and a real eigenfunction for it vanishes at no vertex; for such $n$ let $\phi_n$ be the number of its zeros in the interiors of the edges and $s_n = \phi_n - (n-1)$, the nodal surplus. With cycle rank $\beta = |E| - |V| + 1$, one has $0 \le s_n \le \beta$. Alon, Band and Berkolaiko proved that for each $s$ the proportion of generic indices $n \le N$ with $s_n = s$ converges as $N \to \infty$; let $S$ be a random variable with this limiting distribution. They proved that it is symmetric about $\beta/2$, and binomial with parameters $\beta$ and $1/2$ when the cycles of $\Gamma$ are disjoint, and later that the law below holds for stower graphs (every edge a loop or a pendant edge) and mandarin graphs (two vertices, every edge joining both).

Problem. Is it true that for every sequence $(\Gamma_j)$ of such graphs with $\beta_j \to \infty$, $$\frac{S_j - \beta_j/2}{\sqrt{\operatorname{Var}(S_j)}} \Longrightarrow \mathcal{N}(0,1)$$ in distribution, and $c\beta_j \le \operatorname{Var}(S_j) \le C\beta_j$ for all sufficiently large $j$, with universal constants $c, C > 0$?

Source: Alon, Band and Berkolaiko (2024; arXiv 2021), Conjecture 3.1, after the prediction of Gnutzmann, Smilansky and Weber (2004) that nodal statistics of large graphs are Gaussian.

Imported from github.com/MColbrook/AIM.

Classification

Primary: SP — Spectral Theory
Secondary: MP, PR
Keywords: AIM Open Applied Problems, central limit theorems, nodal counts, quantum graphs

References

@article{alon2024,
  author        = {Alon, Lior and Band, Ram and Berkolaiko, Gregory},
  title         = {Universality of nodal count distribution in large metric graphs},
  journal       = {Exp. Math.},
  volume        = {33},
  number        = {2},
  year          = {2024},
  pages         = {301--335},
  doi           = {10.1080/10586458.2022.2092565},
  eprint        = {2106.06096},
  archivePrefix = {arXiv},
}
@article{gnutzmann2004,
  author  = {Gnutzmann, Sven and Smilansky, Uzy and Weber, Joachim},
  title   = {Nodal counting on quantum graphs},
  journal = {Waves Random Media},
  volume  = {14},
  number  = {1},
  year    = {2004},
  pages   = {S61--S73},
  doi     = {10.1088/0959-7174/14/1/011},
}
@article{alon2018,
  author        = {Alon, Lior and Band, Ram and Berkolaiko, Gregory},
  title         = {Nodal statistics on quantum graphs},
  journal       = {Comm. Math. Phys.},
  volume        = {362},
  number        = {3},
  year          = {2018},
  pages         = {909--948},
  doi           = {10.1007/s00220-018-3111-2},
  eprint        = {1709.10413},
  archivePrefix = {arXiv},
}
@misc{aim-quantum-graph-nodal-clt,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{A universal central limit law for quantum-graph nodal surplus}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/114-quantum-graph-nodal-clt.md},
  note         = {Entry 114 at commit a153f46, 2026-10-13},
}

Resolution claims

A non-Gaussian limit for nodal surplus v1 GitHub ↗

Sidney Holden

For each $m \ge 3$, an explicit connected loop-free quantum graph with cycle rank $\beta_m = 2m - 1$, no vertex of degree two and rationally independent edge lengths, under the standard Kirchhoff Laplacian, whose nodal surplus $S_m$ satisfies $\operatorname{Var}(S_m)/\beta_m \to a_*/2 \in (1/8, 1/4)$ while $(S_m - \beta_m/2)/\sqrt{\operatorname{Var}(S_m)}$ converges to a variance mixture $Z\sqrt{V/a_*}$ of Gaussians with a nonconstant …

Counterexample · not verified · fidelity not established · posted on GitHub 5 Oct 2026

Status history

When	Change	By	Reason
2026-10-14	Needs verification → Solution claimed	—	Recorded here: AIM @ c992980, 114-nodal-surplus-counterexample/PROOF.md.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00321, "A universal central limit law for quantum-graph nodal surplus", clopen.solutions.