  Laplace operators on finite compact metric graphs are considered under the
assumption that matching conditions at graph vertices are of $\delta$ and
$\delta'$ 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.
