In the era of gravitational wave (GW) detection from astrophysical sources by LIGO/VIRGO, it is of great importance to take the quantum gravity effect of graviton-photon (GRAPH) mixing in the cosmic magnetic field to the next level. In this work, we study such an effect and derive for the first time perturbative solutions of the linearized equations of motions of the GRAPH mixing in an expanding universe. In our formalism we take into account all known standard dispersive and coherence breaking effects of photons such as the Faraday effect, the Cotton-Mouton (CM) effect, and the plasma effects in the cosmic magnetic field. Our formalism, applies to a cosmic magnetic field either a uniform or a slowly varying non-homogeneous field of spacetime coordinates with an arbitrary field direction. For binary systems of astrophysical sources of GWs at extragalactic distances with chirp masses $M_\text{CH}$ of a few solar masses, GW present-day frequencies $��_0\simeq 50-700$ Hz, and present-day cosmic magnetic field amplitudes $\bar B_0\simeq 10^{-10}-10^{-6}$ G, the power of electromagnetic radiation generated in the GRAPH mixing at present is substantial and in the range $P_��\simeq 10^6-10^{15}$ (erg/s). On the other hand, the associated power flux $F_��$ is quite faint depending on the source distance with respect to the Earth. Since in the GRAPH mixing the velocities of photons and gravitons are preserved and are equal, this effect is the only one known to us, whose certainty of the contemporary arrival of GWs and electromagnetic radiation at the detector is guaranteed.

Published version in Physical Review D. In the published version the title has been changed by the editors to 'Graviton-Photon mixing' instead of 'GRAPH mixing'. With respect to (v1), in this version (v2), typos have been corrected, new references have been added and an additional section (sec.8) has been added. Typos in Eqs. (40) and (41) have been fixed