Oscillation of graph eigenfunctions

TL;DR

This paper establishes an oscillation formula for the eigenvectors of weighted graph operators, linking the number of sign changes to graph cycles and relating it to existing theorems in spectral graph theory and oscillator networks.

Contribution

It introduces a new oscillation formula for eigenvectors of weighted graph operators, connecting sign changes to graph cycles via a Morse index, with two different proofs and connections to prior theorems.

Findings

The number of sign changes in eigenvectors relates to graph cycles.

The oscillation formula is proven through two different methods.

Connections are made to the nodal--magnetic theorem and oscillator network identities.

Abstract

An oscillation formula is established for the $k$-th eigenvector (assumed to be simple and with non-zero entries) of a weighted graph operator. The formula directly attributes the number of sign changes exceeding $k-1$ to the cycles in the graph, by identifying it as the Morse index of a weighted cycle intersection form introduced in the text. Two proofs are provided for the main result. Additionally, it is related to the nodal--magnetic theorem of Berkolaiko and Colin de Verdi\`ere and to a similar identity of Bronski, DeVille and Ferguson obtained for the linearization of coupled oscillator network equations around a known solution.