Symmetry constraints on phonon dispersion in graphene

We calculate the phonon dispersion for graphene with interactions between the first, second, and third nearest neighbors in the framework of the Born-von Karman model taking into account the constraints imposed by the lattice symmetry. Analytical expressions give the nonzero sound velocity for the out-of-plane (bending) mode. The dispersion of four in-plane modes is determined by coupled equations. Values of the force constants are found in fitting with frequencies at critical points and elastic constants measured on graphite.