We consider Stanley--Reisner rings $k[x_1,\ldots,x_n]/I(\emph{H})$ where $I(\emph{H})$ is the edge ideal associated to some particular classes of hypergraphs. More precisely, we consider hypergraphs that are natural generalizations of line graphs and cycles and compute the graded Betti numbers of their edge ideals. Moreover shellable, sequentially Cohen-Macaulay and Cohen-Macaulay hypercycles are characterized.
