Abstract: Motivated by the application of quantum graphs to model the anomalous Hall effect, we discuss periodic quantum graphs with the vertex coupling which is non-invariant with respect to the time-reversal. We describe an example of such a vertex coupling by analyzing the band spectra of kagome and triangular lattices. Special attention is paid to the asymptotic behavior of the spectral bands in the high-energy regime; we see that the Band-Berkolaiko universality holds as long as the graph edge lengths are incommensurate. Compared with the other examples of quantum graphs with the same coupling condition, we see that the transport properties at the vertex at high energies depend substantially on the network topology, in particular, on the parity of the vertices involved.
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This is joint work, cf. M Baradaran, P Exner, J. Math. Phys. 63 (2022), 083502.