We introduce an upper bound for the photonic Chern number of systems described by either quadratic or linear Hamiltonian-type operators. We use the upper bound to estimate the topological charge of band gaps of nondispersive gyrotropic photonic crystals, without performing any explicit
topological analysis. It is shown that bounds obtained for matrix Hamiltonian operators are generally less tight than those obtained for scalar operators. We extend the analysis to dispersive systems and demonstrate that, in this case, the bound arises from the intrinsic relativistic nature of light.