  We investigate spectral properties of quantum graphs in the form of a
periodic chain of rings with a connecting link between each adjacent pair,
assuming that wave functions at the vertices are matched through conditions
manifestly non-invariant with respect to time reversal. We discuss, in
particular, the high-energy behavior of such systems and the limiting
situations when one of the edges in the elementary cell of such a graph shrinks
to zero. The spectrum depends on the topology and geometry of the graph. The
probability that an energy belongs to the spectrum takes three different values
reflecting the vertex parities and mirror symmetry, and the band patterns are
influenced by commensurability of graph edge lengths.
