Poincar\'e series of some hypergraph algebras

A hypergraph $H=(V,E)$, where $V=\{x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n\}$ and $E\subseteq 2^V$ defines a hypergraph algebra $R_H=k[x_1,…,x_n]/(x_{i_1}… x_{i_k}; {i_1,…,i_k}\in E)$\mathrm{.}Allourhypergraphsare$d$-uniform,i\mathrm{.}e\mathrm{.},$|e_i|=d$forall$e_i\in E$\mathrm{.}WedeterminethePoincar\acute{e}series$P_{R_H}(t)=\sum_{i=1}^\infty\dim_k{\rm Tor}_i^{R_H}(k,k)t^i$ for some hypergraphs generalizing lines, cycles, and stars. We finish by calculating the graded Betti numbers and the Poincar'e series of the graph algebra of the wheel graph.