Hypergraph Characterization of Fusion Rings

We present a correspondence between multiplicity-free, self-dual, fusion rings and a digraph, hypergraph pair $(D,H)$. This correspondence is used to provide a complete characterization of all fusion rings corresponding to graphical properties of $D$. Further, we exploit this correspondence to provide a complete list of all non-isomorphic, self-dual, multiplicity-free fusion rings of rank at most 8.