Dynamics of nodal points and the nodal count on a family of quantum graphs

TL;DR

This paper studies the zeros of eigenfunctions on quantum graphs, deriving formulas that relate zero counts to spectral properties, including explicit formulas for dihedral graphs based solely on edge lengths, revealing zero dynamics.

Contribution

It introduces new formulas linking eigenfunction zeros to spectral data and provides an explicit edge-length-based formula for dihedral graphs, advancing understanding of zero dynamics in quantum graphs.

Findings

Derived formulas connecting zero counts to spectra of graphs and subgraphs

Established an explicit zero count formula for dihedral graphs using only edge lengths

Explained zero dynamics through scattering problem analysis

Abstract

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schr\"odinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.