  Fix an integer $n \geq 1$, and consider the set of all connected finite
simple graphs on $n$ vertices. For each $G$ in this set, let $I(G)$ denote the
edge ideal of $G$ in the polynomial ring $R = K[x_1,\ldots,x_n]$. We initiate a
study of the set $\mathcal{RD}(n) \subseteq \mathbb{N}^2$ consisting of all the
pairs $(r,d)$ where $r = {\rm reg}(R/I(G))$, the Castelnuovo-Mumford
regularity, and $d = {\rm deg} h_{R/I(G)}(t)$, the degree of the
$h$-polynomial, as we vary over all the connected graphs on $n$ vertices. In
particular, we identify sets $A(n)$ and $B(n)$ such that $A(n) \subseteq
\mathcal{RD}(n) \subseteq B(n)$. When we restrict to the family of
Cameron-Walker graphs on $n$ vertices, we can completely characterize all the
possible $(r,d)$.
