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The paper is a continuation of the study started in \cite{Yorzh1}.Schrodinger operators on finite compact metric graphs are considered under theassumption that the matching conditions at the graph vertices are of $\delta$type. Either an infinite series of trace formulae (provided that edgepotentials are infinitely smooth) or a finite number of such formulae (in thecases of $L_1$ and $C^M$ edge potentials) are obtained which link together twodifferent quantum graphs under the assumption that their spectra coincide.Applications are given to the problem of recovering matching conditions for aquantum graph based on its spectrum.

Trace formulae for Schrodinger operators on metric graphs with applications to recovering matching conditions