تقرير
Generating isospectral but not isomorphic quantum graphs
| العنوان: | Generating isospectral but not isomorphic quantum graphs |
|---|---|
| المؤلفون: | Pistol, Mats-Erik |
| سنة النشر: | 2021 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Spectral Theory |
| الوصف: | Quantum graphs are defined by having a Laplacian defined on the edges of a metric graph with boundary conditions on each vertex such that the resulting operator, $\mathbf{L}$, is self-adjoint. We use Neumann boundary conditions although we do a slight excursion into graphs with Dirichlet and $\delta$-type boundary condititons towards the end of the paper. The spectrum of $\mathbf{L}$ does not determine the graph uniquely, that is, there exist non-isomorphic graphs with the same spectra. There are few known examples of pairs of non-isomorphic but isospectral quantum graphs. In this paper we start to correctify this situation by finding hundreds of isospectral sets, using computer algebra. We have found all sets of isospectral but non-isomorphic equilateral connected quantum graphs with at most nine vertices. This includes thirteen isospectral triplets and one isospectral set of four. One of the isospectral triplets involves a loop where we could prove isospectrality. We also present several different combinatorial methods to generate arbitrarily large sets of isospectral graphs, including infinite graphs in different dimensions. As part of this we have found a method to determine if two vertices have the same Titchmarsh-Weyl $M$-function. We give combinatorial methods to generate sets of graphs with arbitrarily large number of vertices with the same $M$-function. We also find several sets of graphs that are isospectral under both Neumann and Dirichlet boundary conditions as well as under more general, $\delta$-type and $\delta'_s$-type, boundary conditions. We discuss the possibilities that our program is incorrect, present our tests and open source it for inspection at http://github.com/meapistol/Spectra-of-graphs Comment: 62 pages, 41 figures. This is an average revision where two more figures have been added involving more general boundary conditions. Typos have been corrected and one reference added. We will generate secular determinants on reasonable requests for the community |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2104.12885 |
| رقم الأكسشن: | edsarx.2104.12885 |
| قاعدة البيانات: | arXiv |
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