The aim of this talk is to illustrate how a nontrivial topology can lead to a rich variety of spectral types. We focus on second-order equations used to describe periodic quantum systems. Such a PDE in a Euclidean space has typically the spectrum which is absolutely continuous, consisting of bands and gaps, the number of the latter being determined by the dimensionality. We are going to show that for analogous second-order operators on metric graphs, many different situations may arise. Using simple examples, we show that the spectrum may then have a pure point or a fractal character, and also that it may have only a finite but nonzero number of open gaps. Furthermore, motivated by recent attempts to model the anomalous Hall effect, we investigate a class of vertex couplings that violate the time reversal invariance. We compare spectra of different lattice graphs with the
simplest coupling of this type and demonstrate that it depends substantially on the parity of the vertices, and discuss some consequences of this property.