  We analyze spectrum of Laplacian supported by a periodic honeycomb lattice
with generally unequal edge lengths and a $\delta$ type coupling in the
vertices. Such a quantum graph has nonempty point spectrum with compactly
supported eigenfunctions provided all the edge lengths are commensurate. We
derive conditions determining the continuous spectral component and show that
existence of gaps may depend on number-theoretic properties of edge lengths
ratios. The case when two of the three lengths coincide is discussed in detail.
