  We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and
generally hyperrectangular) lattice graph with $\delta$ couplings in the
vertices. We formulate sufficient conditions under which the number of gaps is
finite. As the main result, we find a connection between the arrangement of the
gaps and the coefficients in a continued fraction associated with the ratio of
edge lengths of the lattice. This knowledge enables a straightforward
construction of a periodic quantum graph with any required number of spectral
gaps and---to some degree---to control their positions; i.e., to partially
solve the inverse spectral problem.
