We discuss spectral properties quantum graphs with vertex couplings which violate the time-reversal invariance. We focus on the simplest example of such a coupling and address two questions. The first is about bounds on the discrete spectrum, in particular, about the topology which optimizes the ground-state eigenvalue. The other is related to quantum chaos: we are going to demonstrate an example of a graph the properties of which, both in the eigenvalue spacing distribution and the form factor, differ from the random-matrix patterns commonly expected in this situation. The results come from a common work with Ram Band, Divya Goel, Jonathan Rohleder and Aviya Strauss.
