On the inverse spectral problems for quantum graphs

We review some aspects of inverse spectral problems for quantum graphs. Under hypothesis of rational independence of lengths of edges it is possible, thanks to trace formulas, to reconstruct information on compact and non compact graphs from the knowledge, respectively, of the spectrum of Laplacian and of the scattering phase. In the case of Sturm-Liouville operators defined on compact graphs and in general for differential operators on compact star-graphs, unknown potentials can be recovered from the knowledge of the spectrum of operators obtained imposing different boundary conditions.