Let G be a finite simple graph on the vertex set [n] = {1,…, n} and K[x, y] = K[x1,…, xn, y1,…, yn] the polynomial ring in 2n variables over a field K with each degxi=degyj=1\documentclass[12pt]{minimal}
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\begin{document}$\deg x_{i} = \deg y_{j} = 1$\end{document}. The binomial edge ideal of G is the binomial ideal JG ⊂ K[x, y] which is generated by those binomials xiyj − xjyi for which {i, j} is an edge of G. The Hilbert series HK[x,y]/JG(λ)\documentclass[12pt]{minimal}
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\begin{document}$H_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )$\end{document} of K[x, y]/JG is of the form HK[x,y]/JG(λ)=hK[x,y]/JG(λ)/(1−λ)d\documentclass[12pt]{minimal}
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\begin{document}$H_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )/(1 - \lambda )^{d}$\end{document}, where d=dimK[x,y]/JG\documentclass[12pt]{minimal}
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\begin{document}$d = \dim K[\mathbf {x}, \mathbf { y}]/J_{G}$\end{document} and where hK[x,y]/JG(λ)=h0+h1λ+h2λ2+⋯+hsλs\documentclass[12pt]{minimal}
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\begin{document}$h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = h_{0} + h_{1}\lambda + h_{2}\lambda ^{2} + {\cdots } + h_{s}\lambda ^{s}$\end{document} with each hi∈ℤ\documentclass[12pt]{minimal}
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\begin{document}$h_{i} \in \mathbb Z$\end{document} and with hs≠ 0 is the h-polynomial of K[x, y]/JG. It is known that, when K[x, y]/JG is Cohen–Macaulay, one has reg(K[x,y]/JG)=deghK[x,y]/JG(λ)\documentclass[12pt]{minimal}
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\begin{document}$\operatorname {reg}(K[\mathbf {x}, \mathbf {y}]/J_{G}) = \deg h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )$\end{document}, where reg(K[x, y]/JG) is the (Castelnuovo–Mumford) regularity of K[x, y]/JG. In the present paper, given arbitrary integers r and s with 2 ≤ r ≤ s, a finite simple graph G for which reg(K[x, y]/JG) = r and deghK[x,y]/JG(λ)=s\documentclass[12pt]{minimal}
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\begin{document}$\deg h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = s$\end{document} will be constructed.
