Dispersion relations of periodic quantum graphs associated with Archimedean tilings (I)

There are 11 kinds of Archimedean tilings for the plane in total. Applying the Floquet-Bloch theory, we derive the dispersion relations of the periodic quantum graphs associated with a number of Archimedean tilings, namely the triangular tiling (3(6)), the elongated triangular tiling (3(3), 4(2)), the trihexagonal tiling (3, 6, 3, 6) and the truncated square tiling (4,8(2)). The derivation makes use of characteristic functions, with the help of the symbolic software Mathematica. The resulting dispersion relations are surprisingly simple and symmetric. They show that in each case the spectrum is composed of point spectrum and an absolutely continuous spectrum. We further analyzed the structure of absolutely continuous spectra. Our work is motivated by the studies on the periodic quantum graphs associated with hexagonal tiling in Kuchment and Post (2007 Commun. Math. Phys. 275 805-26) and Korotyaev and Lobanov (2007 Ann. Henri Poincare 8 1151-76).