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 $\mathrm{deg}{x}_i=\mathrm{deg}{y}_j=1$. The binomial edge ideal of $G$ is the binomial ideal $J_G\subset K[\mathbf{x},\mathbf{y}]$ which is generated by those binomials $x_i{y}_j-x_j{y}_i$ 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 $h_s\ne 0$ 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 $\mathrm{reg}(K[\mathbf{x},\mathbf{y}]/J_G)=\mathrm{deg}{h}_{K[\mathbf{x},\mathbf{y}]/J_G}(\lambda )$, 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 $2\le r\le s$, a finite simple graph $G$ for which $\mathrm{reg}(K[\mathbf{x},\mathbf{y}]/J_G)=r$ and $\mathrm{deg}{h}_{K[\mathbf{x},\mathbf{y}]/J_G}(\lambda )=s$ will be constructed.