Generating isospectral but not isomorphic quantum graphs

Quantum graphs are defined by having a Laplacian defined on the edges a metric graph with boundary conditions on each vertex such that the resulting operator, $\mathbf{L}$, is self-adjoint. We use Neumann boundary conditions. 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 rectify 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-Weil $M$-function. We give combinatorial methods to generate sets of graphs with arbitrarily large number of vertices with the same $M$-function. 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.