High-energy asymptotics of the spectrum of a periodic square lattice quantum graph

We investigate a periodic quantum graph in the form of a square lattice with a general self-adjoint coupling at the vertices. We analyse the spectrum's 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 like \ O(n^j),\, j=1,0,-1,-2,-3, as the band index n → ∞. The gaps may be of asymptotically constant width or linearly growing with the latter case being generic.