Nodal count of graph eigenfunctions via magnetic perturbation

TL;DR

This paper links the stability of graph Laplacian eigenvalues under magnetic perturbations to the zeros of eigenfunctions, revealing a relationship between nodal surplus and Morse index at zero magnetic field.

Contribution

It introduces a novel connection between eigenvalue stability, magnetic perturbations, and nodal surplus in graph eigenfunctions, providing a new perspective on spectral graph theory.

Findings

Critical point at zero magnetic field corresponds to eigenvalue stability.

Morse index at the critical point equals the nodal surplus.

Eigenfunction zeros relate to the eigenvalue's behavior under magnetic perturbation.

Abstract

We establish a connection between the stability of an eigenvalue under a magnetic perturbation and the number of zeros of the corresponding eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a graph and count the number of edges where the eigenfunction changes sign (has a "zero"). It is known that the $n$-th eigenfunction has $n-1+s$ such zeros, where the "nodal surplus" $s$ is an integer between 0 and the number of cycles on the graph. We then perturb the Laplacian by a weak magnetic field and view the $n$-th eigenvalue as a function of the perturbation. It is shown that this function has a critical point at the zero field and that the Morse index of the critical point is equal to the nodal surplus $s$ of the $n$-th eigenfunction of the unperturbed graph.