Is there a general theorem which gives some information about which influence have the lattice geometry (for example sub-lattice structure, square lattice, honeycomb lattice, lattice symmetries, ...) to the dispersion relation in the low-energy regime?
For example: graphene or magnons in antiferromagnets. Both have a sub-lattice structure and the dispersion relations are linear.
It is true that the lattice structure will determine the symmetries of the band structure. However, this alone cannot determine the low energy excitations because those are determined by the number of mobile electrons in the unit cell, i.e. the Fermi energy (at least in the case of a metal). This is why elements in the periodic table tend to have similar characteristics when going down the columns.
The similarity between the linear dispersion in magnons and graphene may arise from the mathematical similarity in their formulation which comes from their sublattice structure, but it is a a lucky consequence of graphene also having electrons in the $2s^22p^2$ shells be mobile.
