Regularity and h-polynomials of edge ideals

For any two integers d, r ≥ 1, we show that there exists an edge ideal I(G) such that reg (R/I(G)), the Castelnuovo-Mumford regularity of R/I(G), is r, and deg hR/I(G) (t), the degree of the h-polynomial of R/I(G), is d. Additionally, if G is a graph on n vertices, we show that reg (R/I(G)) + deg hR/I(G) (t) ≤ n.