  Let $G$ be a finite simple graph on the vertex set $[n] = \{ 1, \ldots, n \}$
and $K[X, Y] = K[x_1, \ldots, x_n, y_1, \ldots, y_n]$ the polynomial ring in
$2n$ variables over a field $K$ with each $\mathrm{deg} x_i = \mathrm{deg} y_j
= 1$. The binomial edge ideal of $G$ is the binomial ideal $J_G \subset K[X,
Y]$ which is generated by those binomials $x_iy_j - x_jy_i$ for which $\{i,
j\}$ is an edge of $G$. The Hilbert series $H_{K[X, Y]/J_G}(\lambda)$ of $K[X,
Y]/J_G$ is of the form $H_{K[X, Y]/J_G}(\lambda) = h_{K[X, Y]/J_G}(\lambda)/(1
- \lambda)^d$, where $d = \mathrm{dim} K[X, Y]/J_G$ and where $h_{K[X,
Y]/J_G}(\lambda) = h_0 + h_1\lambda + h_2\lambda^2 + \cdots + h_s\lambda^s$
with each $h_i \in \mathbb{Z}$ and with $h_s \neq 0$ is the $h$-polynomial of
$K[X, Y]/J_G$. It is known that, when $K[X, Y]/J_G$ is Cohen-Macaulay, one has
$\mathrm{reg}(K[X, Y]/J_G) = \mathrm{deg} h_{K[X, Y]/J_G}(\lambda)$, where $
\mathrm{reg}(K[X, Y]/J_G)$ is the (Castelnuovo-Mumford) regularity of $K[X,
Y]/J_G$. In the present paper, given arbitrary integers $r$ and $s$ with $2
\leq r \leq s$, a finite simple graph $G$ for which $\mathrm{reg}(K[X, Y]/J_G)
= r$ and $\mathrm{deg} h_{K[X, Y]/J_G}(\lambda) = s$ will be constructed.
