Abstract
The honeycomb supersolid state is predicted to form in a dipolar Bose-Einstein condensate with a planar confining potential. Our results for its excitation spectrum reveal the gapless bands and the emergence of Dirac points at the Brillouin zone edge, manifesting as points where the second sound and transverse sound bands touch. The honeycomb supersolid has three sound speeds that we connect to its elastic parameters through hydrodynamic theory. From this analysis we find conditions where a shear instability occurs as the honeycomb rigidity disappears. This gives insight into the nonequilibrium dynamics following an interaction quench, where the honeycomb pattern melts and different crystal orders emerge.