Spectrally grown graphs

TL;DR

This paper introduces a method for designing quantum graphs with desired spectral properties by evolving graphs from a starting point, demonstrating the approach's effectiveness through experiments and open-sourcing the software.

Contribution

The paper presents a novel spectrally grown graph method for inverse spectral problems in quantum graphs, combining analytical and computational techniques.

Findings

The method can find graphs with spectra close to prescribed values.

Spectrally grown graphs can be evolved to meet specific spectral criteria.

The approach enables new conjectures on quantum graph spectra.

Abstract

Quantum graphs have attracted attention from mathematicians for some time. A quantum graph is defined by having a Laplacian on each edge of a metric graph and imposing boundary conditions at the vertices to get an eigenvalue problem. A problem studying such quantum graphs is that the spectrum is timeconsuming to compute by hand and the inverse problem of finding a quantum graph having a specified spectrum is difficult. We solve the forward problem, to find the eigenvalues, using a previously developed computer program. We obtain all eigenvalues analytically for not too big graphs that have rationally dependent edges. We solve the inverse problem using "spectrally grown graphs". The spectrally grown graphs are evolved from a starting (parent) graph such that the child graphs have eigenvalues are close to some criterion. Our experiments show that the method works and we can usually find graphs having spectra which are numerically close to a prescribed spectrum. There are naturally exceptions, such as if no graph has the prescribed spectrum. The selection criteria (goals) strongly influence the shape of the evolved graphs. Our experiments allows us to make new conjectures concerning the spectra of quantum graphs. We open-source our software at https://github.com/meapistol/Spectra-of-graphs.