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We investigate spectral properties of quantum graphs in the form of aperiodic chain of rings with a connecting link between each adjacent pair,assuming that wave functions at the vertices are matched through conditionsmanifestly non-invariant with respect to time reversal. We discuss, inparticular, the high-energy behavior of such systems and the limitingsituations when one of the edges in the elementary cell of such a graph shrinksto zero. The spectrum depends on the topology and geometry of the graph. Theprobability that an energy belongs to the spectrum takes three different valuesreflecting the vertex parities and mirror symmetry, and the band patterns areinfluenced by commensurability of graph edge lengths.

Spectrum of periodic chain graphs with time-reversal non-invariant vertex coupling