Magnetic ring chains with vertex coupling of a preferred orientation

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 edges are incommensurate. The influences probability energy belongs spectrum in tight-chain case, also it can turn some bands into infinitely degenerate eigenvalues.