Can One Hear the Shape of a Network?

Already in 1892, in a book about spectrocospy, A.Shuster raised the question ”... how to find a shape of a bell by means of the sounds which it is capable of sending out.” [15] The inverse spectral problem hidden behind the physical context was first formulated in a mathe-matical setting by H.Weyl indirectly in 1911 and by S. Bochner in 1950, cf. [15]. Finally, in 1966, M.Kac published the famous paper [16] intitled ”Can one hear the shape of a drum?” After a separation ansatz of the wave equation, the problem reads in mathematical terms as follows. Suppose that two bounded domains Ω1 and Ω2 in are isospectral i.e. the spectra of their Laplacian under the homogeneous Dirichlet boundary condition (D) or under the Neumann boundary condition (N) coincide counting multiplicities. Does this imply that Ω1 and Ω2 are isometric, i.e. do they coincide up to rotations, reflec-tions or translations?