Spectrally grown graphs
Mats-Erik Pistol, Pavel Kurasov

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…
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Taxonomy
TopicsSpectral Theory in Mathematical Physics · Graph theory and applications · Quantum Computing Algorithms and Architecture
