If gravitons do exist, will an anti-graviton repel matter(anti-gravity)? Also, can we, reasonably, assume that the graviton does actually exist?
Consider the yukawa interation, the force is always attractive no matter the nucleons are particle or anti-particle. EM interation is a special case, because particle and anti-particle has opposite electron charge. But other charge is not the same, in your case, the anti-particle has positive mass, but not negative mass, so no anti-gravity.
This is a strange question, or better this question asks for a strange answer. Because an anti-graviton could only be an antigraviton if it has opposite energy. So negative mass/energy. This assuming it generates opposite gravitation. But if the graviton does exist, will it still have a positive mass opposite particle? How would such a particle differ? We cannot assume the graviton exists. If you really have to pick yes or no maybe yes would be smarter. But gravitons might not exist. I would be interested in seeing someone with some real physics knowledge answering this question.
Without actually going into the details of the physics of gravitons and quantum gravity, there is already a flaw in your logic here. You are assuming that if there is a force carrier particle such as a gravitons, then automatically it implies that there will be an "anti-gravitons". This is not true in every cases. A photon, which is the force carrier for EM interactions, is its own anti particle! This means that just because we have a particulate carrier, does not automatically imply that an "anti particle" of an opposite kind has to exist. Zz.
There's a nice mathematical explanation to the existence of antiparticles.It involves involuted associative algebras.For example,the complex numbers form such an algebra and the complex scalar field,which is an element of such an algebra,describes,after quantization both particles & antiparticles. See if the electromagnetic field [itex] A_{\mu}(x) [/itex] and the gravity field (well,either one of the 3 possible fields describing it) could form such an algebra. Daniel.
But then could anti-gravity work through gravitons if you have negative mass(made of negative energy?). Opposite charges idea.
There's no such thing as negative mass.You can have imaginary mass emerging when making the unitary infinite dimensional reps of the restricted PoincarĂ© group,the so-called tachyonic reps,but that's something nonphysical. Daniel.