  Quantum graphs are defined by having a Laplacian defined on the edges a
metric graph and having boundary conditions on each vertex such that the
resulting operator, L, is self-adjoint. We use Neumann boundary conditions at
the vertices. The spectrum of L does not determine the graph uniquely, that is,
there exist non-isomorphic graphs with the same spectra. This is similar to the
case with a Laplacian defined on a two-dimensional domain, using Dirichlet or
Neumann boundary conditions, where there exist non-isomorphic domains with the
same spectra. There are few known examples of pairs of non-isomorphic but
isospectral quantum graphs and such graphs must have rationally dependent edge
lengths. We have found all pairs of isospectral but non-isomorphic connected
quantum graphs with at most seven vertices where all edges have equal length.
Our pairs include graphs with different Euler characteristic as well as pairs
differing in the number of vertices. We have also investigated tree graphs and
decorated loops. We find two isospectral triplets as well as a few very simple
examples of isospectral pairs that can be checked by hand. More than 70
isospectral sets have been found. We have done this using computer algebra
where we get all eigenvalues analytically for graphs with rationally dependent
edge lengths. We discuss the possibilities that our program is incorrect,
present our tests and open source it for inspection at this
$\href{http://github.com/meapistol/Spectra-of-graphs}{url}$.
