We show that the sets of classical and non-signaling correlations, and the maximum quantum value for the correlations of any experiment can be completely characterized through three numbers associated to the graph in which vertices are the elementary propositions the experiment can be decomposed into and edges join exclusive ones. This interface between graph theory and physics provides a new perspective to interpret and reprove known results, and a tool to single out interesting experiments. We illustrate the power of the method by constructing the simplest (in some sense) experiment with a quantum-classical gap, and the simplest bipartite Bell inequality showing "fully non-local correlations" since no non-signalling theory can give larger-than-quantum correlations.
