Kagome network with vertex coupling of a preferred orientation

We investigate spectral properties of periodic quantum graphs in the form a kagome or triangular lattice situation when condition matching wave functions at vertices is chosen particular violating time-reversal invariance. The positive spectrum consists infinite number bands, some which may be flat; negative one has most three and two respectively. example shows that even with such an uncommon vertex coupling universality hold: if its edges are incommensurate, probability randomly contained $\approx 0.639$.