  Assuming that the short gamma-ray burst detected by the Fermi Gamma-Ray Space
Telescope about 0.4 seconds after the gravitational waves observed by the LIGO
and VIRGO Collaborations originated from the same black hole merger event, we
perform a model-independent analysis of different quantum gravity scenarios
based on (modified) dispersion relations (typical of quantum gravity models)
for the graviton and the photon. We find that only scenarios where at least one
of the two particles is luminal (the other being sub- or super-luminal) are
allowed, while scenarios where none of the two particles is luminal are ruled
out. Moreover, the physical request of having acceptable values for the quantum
gravity scale imposes stringent bounds on the difference between the velocities
of electromagnetic and gravitational waves, much more stringent than any
previously known bound.
