Regularity and $h$-polynomials of edge ideals

For any two integers $d,r\ge 1$, we show that there exists an edge ideal $I(G)$ such that the $\mathrm{r}\mathrm{e}\mathrm{g}(R\mathrm{/}I(G))$, the Castelnuovo-Mumford regularity of $R\mathrm{/}I(G)$, is $r$, and $\mathrm{d}\mathrm{e}\mathrm{g}(h_{R\mathrm{/}I(G)}(t))$, the degree of the $h$-polynomial of $R\mathrm{/}I(G)$, is $d$. Additionally, if $G$ is a graph on $n$ vertices, we show that ${\rm reg}\left(R/I(G)\right) + {\rm deg} (h_{R/I(G)}(t)) \leq n$.