Generating isospectral but not isomorphic quantum graphs

TL;DR

This paper systematically finds and analyzes hundreds of pairs and sets of non-isomorphic quantum graphs that share the same spectrum, expanding understanding of spectral graph theory and quantum graph isospectrality.

Contribution

It provides the first comprehensive classification of isospectral non-isomorphic quantum graphs with up to nine vertices using computer algebra.

Findings

Identified all isospectral non-isomorphic equilateral connected quantum graphs with ≤9 vertices.

Discovered multiple triplets and quadruples of isospectral graphs, including a proven loop triplet.

Developed combinatorial methods to generate large sets of isospectral graphs, including infinite graphs.

Abstract

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 more general, permutation invariant, boundary conditions. This necessitates a study of eigenvalue zero where we prove several results. 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