Some algebraic properties of hypergraphs

We consider Stanley–Reisner rings $k[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]\mathrm{/}I(\text{mc}H)$ where $I(\text{mc}H)$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize upon some known results about chordal graphs and study a weak form of shellability.