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