  Spectral problems are considered generated by the Sturm-Liouville equation on
connected simple equilateral graphs with the Neumann and Dirichlet boundary
conditions at the pendant vertices and continuity and Kirchhoff's conditions at
the interior vertices. We highlight the cases where the first and the second
terms of the asymptotics of the eigenvalues uniquely determine the shape of the
graph or of its interior subgraph.
