  We present full description of spectra for a Hamiltonian defined on periodic
hexagonal elastic lattices. These continua are constructed out of
Euler-Bernoulli beams, each governed by a scalar-valued self-adjoint operator,
which is also known as the fourth order Schr\"{o}dinger operator, equipped with
a real periodic symmetric potential. In contrast to the second order
Schr\"{o}dinger operator commonly applied in quantum graph literature, here
vertex matching conditions encode geometry of the underlying graph by their
dependence on angles at which edges are met. We show that for a special equal
angle lattice, known as graphene, dispersion relation has a similar structure
as reported for the periodic second order Schr\"{o}dinger operator on hexagonal
lattices. This property is then further utilized to prove existence of singular
Dirac points. We further discuss reducibility of Fermi surface at uncountably
many low-energy levels for this special lattice. Applying perturbation
analysis, we extend the developed theory to derive dispersion relation for
angle-perturbed Hamiltonian of hexagonal lattices in a geometric neighborhood
of graphene. In these graphs, unlike graphene, dispersion relation is not
splitted into purely energy and quasimomentum dependent terms, however singular
Dirac points exist similar to the graphene case.
