  We obtain the mass parameter for a class of static and spherically symmetric
regular black holes (BHs) (namely Bardeen, Hayward and
Ay\'{o}n-Beato-Garc\'{i}a BHs) which are solutions of Einstein's field
equations coupled to nonlinear electrodynamics (NED) in terms of redshifts and
blueshifts of photons emitted by geodesic particles (for instance, stars)
orbiting around these BHs. The motion of photons is not governed by null
geodesics for these type of spacetime geometries which reflects the direct
effects of the electrodynamic nonlinearities in the photon motion; hence, an
effective geometry needs to be constructed to study null trajectories [Phys.
Rev. D61, 045001 (2000)]. To achieve the above, we first study the constants of
motion from the analysis of the motion of both geodesic particles moving in
stable circular orbits and photons ejected from them and reaching a distant
observer (or detector) in the equatorial plane for the above mentioned regular
BHs. The relationship between red/blueshifts of photons and the regular BH
observables is presented. We also numerically find the bounds on the photon
shifts for these regular BH cases.

General Relativity and Quantum Cosmology