Inverse Sturm-Liouville problem of recovering coefficients in boundary conditions

The spectral Sturm-Liouville problem on an equilateral star graph of three edges with the standard conditions at the interior vertex and the Robin conditions at the pendant vertices is considered. It is shown that under the conditions of zero potentials on the edges the asymptotics of the eigenvalues determine the shape of the graph and the constants in the Roben conditions. This method can be applied for the case of more complicated equilateral simple connected graphs.