We analyze a square lattice graph in magnetic field assuming that the vertex coupling is of particular type violating time reversal invariance. Calculating spectrum numerically for rational values flux per plaquette we show how two effects compete; at high energies it which dominates restoring asymptotically familiar Hofstadter’s butterfly pattern.

We discuss spectral properties of an periodic quantum graph consisting array rings coupled either tightly or loosely through connecting links, assuming that the vertex coupling is manifestly non-invariant with respect to time reversal and a homogeneous magnetic field perpendicular plane present. It shown parity determines behavior at high energies Band-Berkolaiko universality holds whenever edg...