Does the existence (now proved) of gravitational waves imply the existence of Gravitons? I studied the theoretical part about the Gravitational waves in General Relativity (linearization of gravity and small perturbations of the metric and so on).
But I was wondering about: since electromagnetic radiation is composed/carried by Photons (or better:the EM force), shall gravitational waves be composed/carried by Gravitons? 
In the end:do gravitational waves imply the existence of gravitons? Or it's something unrelated and off topic? 
 A: 
In the end: do gravitational waves implies the existence of gravitons? Or it's something unrelated and off topic? 

The concept of gravitons arises from a quantized gravity, and quantization of gravity is still an effective theory since it has infinities that have not been renormalizable or containable in some sense in the effective theories used on trust, that the matter will be settled.
String theories do have mathematically consistent quantization of gravity, and gravitons, but they are not at a research point to propose a standard model that will describe all existing data and predict new phenomena.
Thus the graviton is an expected new particle, but even its theoretical existence is precarious, until a definitive quantization of gravity.
Experimentally, when the BICEP2 experiment was thought to have detected gravitational waves from the imprint of their polarization on the cosmic microwave background, the footprint of gravitons , i.e. quantized energy transfer, was thought to have been detected.

Craig Hogan, director of the Fermilab Center for Particle Astrophysics, told physicsworld.com that "If it's confirmed, it is truly profound – the first direct evidence not only for inflation, but of a quantum behaviour of space and time. The image of polarization is a relic imprint of roughly a single quanta of graviton action."

Unfortunately the signal could not be separated from scatterings off space dust. A new experiment with higher resolution, BICEP3 has started data taking and we have to wait and see.
A: No.
To prove that gravity uses gravitons requires experiments that have not yet been done, and further more are likely very hard to do. 
The experiment discussed here is a proposal and it's still a large factor away from getting to any sort of quantum level.
A: While the detection of gravitational waves does not directly imply that gravitational force exists in the realm of the particle-wave duality, it does provide definitive link to the existence of the graviton. I sat in the assembly hall at UMCP during the celebration of the gravitational wave discovery and this very question was asked to the panel. Their response was poignant and quite direct:

“We of course know that the graviton exists, so we did not need this detection to add validity to the claims. The burden of proof is a heavy one and one that we simply do not have the current means to tackle.”

A: No, think of gravitational waves as normal ocean waves. When you see a wave gliding across the sea, it's merely the result of a Force acting on that sea, not the seas creating the wave. There may be Gravitins, but they can't create a wave within themselves.
