In this paper a new class of isospectral graphs is presented. These graphs are isospectral with respect to both the normalized 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.