Degenerate band edges in periodic quantum graphs

Edges of bands of continuous spectrum of periodic structures arise as maxima and minima of the dispersion relation of their Floquet--Bloch transform. It is often assumed that the extrema generating the band edges are non-degenerate. This paper constructs a family of examples of $\mathbb{Z}^3$-periodic quantum graphs where the non-degeneracy assumption fails: the maximum of the first band is achieved along an algebraic curve of co-dimension 2. The example is robust with respect to perturbations of edge lengths, vertex conditions and edge potentials. The simple idea behind the construction allows generalizations to more complicated graphs and lattice dimensions. The curves along which extrema are achieved have a natural interpretation as moduli spaces of planar polygons.