  We propose a hypergraph expansion which facilitates the direct treatment of
quantum spin models with many-site interactions via perturbative linked cluster
expansions. The main idea is to generate all relevant subclusters and sort them
into equivalence classes essentially governed by hypergraph isomorphism.
Concretely, a reduced K\"onig representation of the hypergraphs is used to make
the equivalence relation accessible by graph isomorphism. During this procedure
we determine the embedding factor for each equivalence class, which is used in
the final resummation in order to obtain the final result. As an instructive
example we calculate the ground-state energy and a particular excitation gap of
the plaquette Ising model in a transverse field on the three-dimensional cubic
lattice.
