Let $G$ be a finite simple graph on the vertex set $[n]=\{1,\dots,n\}$ and $K[\mathbf{x},\mathbf{y}]=K[x_1,\dots,x_n,y_1,\dots,y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $\deg x_i=\deg y_j=1$. The binomial edge ideal of $G$ is the binomial ideal $J_G \subset K[\mathbf{x},\mathbf{y}]$ which is generated by those binomials $x_iy_j-x_jy_i$ for which $\{i, j\}$ is an edge of $G$. The Hilbert series $H_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )$ of $K[\mathbf{x},\mathbf{y}]/J_G$ is of the form $H_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )/(1 - \lambda )^{d} $ where $d = \dim K[\mathbf {x}, \mathbf { y}]/J_{G} $ and where $h_{K[\mathbf {x}, \mathbf {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 $h_s\not=0$ is the $h$-polynomial of $K[\mathbf{x},\mathbf{y}]/J_G$. It is known that, when $K[\mathbf{x},\mathbf{y}]/J_G$ is Cohen–Macaulay, one has $\operatorname {reg}(K[\mathbf {x}, \mathbf {y}]/J_{G}) = \deg h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )$, where $\mathrm{reg}(K[\mathbf{x},\mathbf{y}]/J_G)$ is the (Castelnuovo–Mumford) regularity of $K[\mathbf{x},\mathbf{y}]/J_G$. In the present paper, given arbitrary integers $r$ and $s$ with $2\leq r\leq s$, a finite simple graph $G$ for which $\mathrm{reg}(K[\mathbf{x},\mathbf{y}]/J_G)=r$ and $\deg h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = s$ will be constructed.