Isospectrality for graph Laplacians under the change of coupling at graph vertices

Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and ${\delta }^{\mathrm{\prime }}$ types. An infinite series of trace formulae is obtained which link together two different quantum graphs under the assumption that their spectra coincide. The general case of graph Schrodinger operators is also considered, yielding the first trace formula. Tightness of results obtained under no additional restrictions on edge lengths is demonstrated by an example. Further examples are scrutinized when edge lengths are assumed to be rationally independent. In all but one of these impossibility of isospectral configurations is ascertained.