  We analyze band spectrum of the periodic quantum graph in the form of a chain
of rings connected by line segments with the vertex coupling which violates the
time reversal invariance, interpolating between the $\delta$ coupling and the
one determined by a simple circulant matrix. We find that flat bands are
generically absent and that the negative spectrum is nonempty even for
interpolation with a non-attractive $\delta$ coupling; we also determine the
high-energy asymptotic behavior of the bands.
