  We discuss spectral properties of an periodic quantum graph consisting of an
array of rings coupled either tightly or loosely through connecting links,
assuming that the vertex coupling is manifestly non-invariant with respect to
the time reversal and a homogeneous magnetic field perpendicular to the graph
plane is present. It is shown that the vertex parity determines the spectral
behavior at high energies and the Band-Berkolaiko universality holds whenever
the edges are incommensurate. The magnetic field influences the probability
that an energy belongs to the spectrum in the tight-chain case, and also it can
turn some spectral bands into infinitely degenerate eigenvalues.
