  A new class of isospectral graphs is presented. These graphs are isospectral
with respect to both the normalised Laplacian on the discrete graph and the
standard differential Laplacian on the corresponding metric graph. The new
class of graphs is obtained by gluing together subgraphs with the Steklov maps
possessing special properties. It turns out that isospectrality is related to
the degeneracy of the Steklov eigenvalues.
