Asymptotically isospectral periodic graphs

We consider discrete Laplace operators on periodic graphs. Their spectrum consists of a finite number of bands. We perturb a periodic graph by adding long edges, i.e., edges connecting distant vertices (by periodically way) and show that if the length of the added edges tends to infinity, then the perturbed graph is asymptotically isospectral to some periodic graph of a larger dimension but without long edges. One of the simplest examples of such asymptotically isospectral periodic graphs is the square lattice perturbed by long edges and the cubic lattice.