What is the interaction between a graviton and antimatter? If gravitons were proved to exist, what would the interaction between a graviton and antimatter be?
 A: In the effective quantized gravity theories ( effective means not rigorously derived) the graviton is a zero mass , spin two particle that is its own antiparticle. As said in the comments to your question you are really asking "how gravity affects antimatter."
In the standard model of particle physics matter and antimatter are described with four vectors whose "length" is the mass of the particle or antiparticle.The question of how antiparticles react to gravity is explored experimentally at CERN

We learn it at high school: release two objects of different mass in the absence of friction forces and they fall down at the same rate in Earth’s gravity. What we haven’t learned, because it hasn’t been directly measured in experiments, is whether antimatter falls down at the same rate as ordinary matter or if it might behave differently. Two new experiments at CERN, ALPHA-g and GBAR, have now started their journey towards answering this question.

We have to wait for the data to see if there is a difference.
