Gravitons are hypothetical particles that are supposed to mediate the gravitational interaction, just like photons mediate the electromagnetic interaction. In theories of quantum gravity, such as string theory, gravitons are predicted to be massless and travel at the speed of light. This is because gravity has a very long range and appears to propagate at the speed of light, as observed by gravitational waves.
However, there are some alternative theories, collectively known as massive gravity, that propose that gravitons have a very small but non-zero mass and move slower than the speed of light. This would imply that gravity is not a fundamental force but an emergent phenomenon, and that it has a finite range and a variable strength depending on the distance.
The existence of massive gravitons would have observable consequences, such as modifying the gravitational lensing of distant galaxies, altering the polarization of gravitational waves, and causing a tiny delay in the arrival time of photons from distant sources (1). However, none of these effects have been detected so far, and the current experimental limits on the graviton mass are very stringent. For example, observations of weak lensing of distant galaxies have placed an upper bound on the graviton mass of less than 6 × 10−32 eV/c2, which is equivalent to about 10−55 kg.
Therefore, based on the available evidence, it is more likely that gravitons are massless and move at the speed of light, as predicted by general relativity. However, this does not rule out the possibility of massive gravitons entirely, and more precise measurements and experiments are needed to test this hypothesis further.
(1) Photons could reveal 'massive gravity,' new theory suggests. https://www.livescience.com/graviton-photon-interaction-hunting-machine.html.