Periodic spectrum of $n$-cubic quantun graphs

We study the spectrum of some periodic differential operators, in particular the periodic Schr\"{o}dinger operator acting on infinite $n$-cubic graphs. Using Floquet-Bloch theory, we derive and analyze on the dispersion relations of the periodic quantum graph generated by 2-dimensional rectangles, and also $n$-cubes. Our proof is analytic. These dispersion relations define the spectra of the associated periodic operator, thus facilitating further analysis of the spectra.