We present full description of spectra for elastic beam Hamiltonian defined on periodic hexagonal lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar valued self-adjoint fourth-order operator equipped with a real periodic symmetric potential. Compared to the Schrödinger operator commonly applied in quantum graph literature, here vertex matching conditions encode geometry of the 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 Schrödinger operator on periodic hexagonal lattices. This property is then further utilized to prove existence of singular Dirac points. We next discuss the role of the potential on reducibility of Fermi surface at uncountably many low-energy levels for this special lattice. Applying perturbation analysis, the developed theory is extended to derive dispersion relation for angle-perturbed Hamiltonian of lattices in a geometric-neighborhood of graphene. In these graphs, unlike graphene, dispersion relation is not splitted into purely energy and quasimomentum dependent terms, but up to some quantifiable accuracy, singular Dirac points exist at the same points as the graphene case.