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, $L$, is self-adjoint. We use Neumann boundary conditions. The spectrum of $L$ 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. Using computer algebra we have found all pairs 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 two different combinatorial method to generate arbitrarily large sets of isospectral graphs, where one method involves star graphs having pumpkin graphs as leaves. We discuss the possibilities that our program is incorrect, present our tests and open source it for inspection at github.com/meapistol/Spectra-of-graphs.Comment: 35 pages, 25 figures. Typos corrected. This is a minor revision with only one figure added. We have here included sets of isospectral graphs with ten vertices and a few secular determinants have been included, notably for the complete graph. We will generate secular determinants on reasonable requests for the communit