Morse theory for discrete magnetic operators and nodal count distribution for graphs

TL;DR

This paper applies Morse theory to discrete magnetic operators on graphs, revealing new geometric insights into eigenvector sign changes and analyzing the distribution of nodal counts, with implications for spectral graph theory.

Contribution

It extends Morse theoretic analysis to magnetic perturbations of discrete Schrödinger operators, linking critical points to eigenvector properties and nodal surplus distributions.

Findings

Critical points form nondegenerate submanifolds related to planar linkages.

Average nodal surplus distribution is binomial under certain conditions.

Conjecture: nodal surplus distribution approaches Gaussian as graph complexity increases.

Abstract

Given a discrete Schr\"odinger operator $h$ on a finite connected graph $G$ of $n$ vertices, the nodal count $\phi(h,k)$ denotes the number of edges on which the $k$-th eigenvector changes sign. A {\em signing} $h'$ of $h$ is any real symmetric matrix constructed by changing the sign of some off-diagonal entries of $h$, and its nodal count is defined according to the signing. The set of signings of $h$ lie in a naturally defined torus $\mathbb{T}_h$ of ``magnetic perturbations" of $h$. G. Berkolaiko discovered that every signing $h'$ of $h$ is a critical point of every eigenvalue $\lambda_k:\mathbb{T}_h \to \mathbb{R}$, with Morse index equal to the nodal surplus. We add further Morse theoretic information to this result. We show if $h_{\alpha} \in \mathbb{T}_h$ is a critical point of $\lambda_k$ and the eigenvector vanishes at a single vertex $v$ of degree $d$, then the critical point lies in a nondegenerate critical submanifold of dimension $d+n-4$, closely related to the configuration space of a planar linkage. We compute its Morse index in terms of spectral data. The average nodal surplus distribution is the distribution of values of $\phi(h',k)-(k-1)$, averaged over all signings $h'$ of $h$. If all critical points correspond to simple eigenvalues with nowhere-vanishing eigenvectors, then the average nodal surplus distribution is binomial. In general, we conjecture that the nodal surplus distribution converges to a Gaussian in a CLT fashion as the first Betti number of $G$ goes to infinity.