Hilbert series and suspensions of graphs

We explore the relationship between the Hilbert series of the edge ideal I of a graph and the combinatorial invariants of the graph, with a focus on identifying relationships between entries of the h-vector of R/I and graph properties. When the graph is a suspension, and thus Cohen-Macaulay with positive entries in the h-vector, we show that those entries are equal to the entries of the f-vector of the Stanley-Reisner complex of the induced subgraph on the vertices of degree at least 2.