Regular Edges, Matchings and Hilbert Series

When $I$ is the edge ideal of a graph $G$, we use combinatorial properities, particularly Property $P$ on connectivity of neighbors of an edge, to classify when a binomial sum of vertices is a regular element on $R/I(G)$. Under a mild separability assumption, we identify when such elements can be combined to form a regular sequence. Using these regular sequences, we show that the Hilbert series and corresponding $h$-vector can be calculated from a related graph using a simplified calculation on the $f$-vector, or independence vector, of the related graph. In the case when the graph is Cohen-Macaulay with a perfect matching of regular edges satisfying the separability criterion, the $h$-vector of $R/I(G)$ will be precisely the $f$-vector of the Stanley-Reisner complex of a graph with half as many vertices as $G$.