Joseph P. Brennan, Kyle Trainor:

Given a graph with loops or a hypergraph, this paper demonstrates that the Hilbert series of the associated graph or hypergraph ring is determined by the Hilbert series of the graph or hypergraph rings associated to the contraction, deletion, and extraction of an edge of the original graph or hypergraph. This enables inductive procedures to compute the Hilbert series of graph and hypergraph rings.