  We consider a quantum mechanical particle living on a graph and discuss the
behaviour of its wavefunction at graph vertices. In addition to the standard
(or delta type) boundary conditions with continuous wavefunctions, we
investigate two types of a singular coupling which are analogous to the delta'
interaction and its symmetrized version for particle on a line. We show that
these couplings can be used to model graph superlattices in which point
junctions are replaced by complicated geometric scatterers. We also discuss the
band spectra for rectangular lattices with the mentioned couplings. We show
that they roughly correspond to their Kronig-Penney analogues: the delta'
lattices have bands whose widths are asymptotically bounded and do not approach
zero, while the delta lattice gap widths are bounded. However, if the
lattice-spacing ratio is an irrational number badly approximable by rationals,
and the delta coupling constant is small enough, the delta lattice has no ggaps
above the threshold of the spectrum. On the other hand, infinitely many gaps
emerge above a critical value of the coupling constant; for almost all ratios
this value is zero.
