Isospectrality for graph Laplacians under the change of coupling at graph vertices: necessary and sufficient conditions

Laplace 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. Assuming rational independence of edge lengths, necessary and sufficient conditions of isospectrality of two Laplacians defined on the same graph are derived and scrutinized. It is proved that the spectrum of a graph Laplacian uniquely determines matching conditions for “almost all” graphs.