We developed a method which enables one to construct isospectral objects, such as quantum graphs and drums. This method is based on some basic representation theory arguments. For an explanation of the method and some neat examples, you might want to have a look at the isospectral construction method for drums (Oxford University SIAM Student Chapter Conference, February 2011) or the same construction demonstrated on quantum graphs (satellite workshop in Edinburgh belonging to the semester on Inverse Problems at the Isaac Newton Institute, October 2011).
Quantum graphs can be extended to scattering systems when they are connected by leads to infinity. We can show that for certain extensions, the scattering matrices of isospectral graphs are conjugate to each other and their poles distributions are therefore identical. I discussed the issue of scattering from isosepctral graphs in a talk given in East Midlands Mathematical Physics Seminars, Loughborough University (December 2010). You can also watch this talk as was given in the Isaac Newton Institute Workshop – Analysis on Graphs and its Applications, Cambridge (June 2010).