  We consider discrete Schr\"odinger operators with periodic potentials on
periodic graphs. Their spectra consist of a finite number of bands. We perturb
a periodic graph by adding edges in a periodic way (without changing the vertex
set) and show that if the added edges are long enough, then the perturbed graph
is asymptotically isospectral to some periodic graph of a higher dimension but
without long edges. We also obtain a criterion for the perturbed graph to be
not only asymptotically isospectral but just isospectral to this higher
dimensional periodic graph. One of the simplest examples of such asymptotically
isospectral periodic graphs is the square lattice perturbed by long edges and
the cubic lattice. We also get asymptotics of the endpoints of the spectral
bands for the Schr\"odinger operator on the perturbed graph as the length of
the added edges tends to infinity.
