  We investigate a periodic quantum graph in form of a square lattice with a
general self-adjoint coupling at the vertices. We analyze the spectrum, in
particular, its high-energy behaviour. Depending on the coupling type, bands
and gaps have different asymptotics. Bands may be flat even if the edges are
coupled, and non-flat band widths may behave as $\mathcal{O}(n^j),\,
j=1,0,-1,-2,-3$, as the band index $n\to\infty$. The gaps may be of
asymptotically constant width or linearly growing with the latter case being
generic.
