Edge switching transformations of quantum graphs

TL;DR

This paper investigates how basic graph transformations, specifically edge switches, affect the spectra of quantum graphs, showing that spectra are interlaced after such transformations, with implications for spectral analysis.

Contribution

It demonstrates that edge switching transformations cause spectra of quantum graphs to be interlaced, providing a new understanding of spectral stability under graph modifications.

Findings

Spectra are interlaced after edge switch transformations.

Spectral bounds are established: $E_{n-2}\le ilde{E}_n \le E_{n+2}$.

Proofs utilize discrete analogs of quantum graphs.

Abstract

Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schr\"odinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by $\{ E_n\}_{n=1}^{\infty}$ and $\{\widetilde E_n\}_{n=1}^{\infty}$ correspondingly, are level-2 interlaced, so that $E_{n-2}\le \widetilde E_n\le E_{n+2}$. The proofs are guided by considerations of the quantum graphs' discrete analogs.