Regularity and h-polynomials of binomial edge ideals

Let $G$ be a finite simple graph on the vertex set $[n]=\{1,\dots ,n\}$ and $K[X,Y]=K[x_1,\dots ,x_n,y_1,\dots ,y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $$\mathrm{d}\mathrm{e}\mathrm{g}$ x_i = $\mathrm{d}\mathrm{e}\mathrm{g}$ y_j = 1$\mathrm{.}Thebinomialedgeidealof$G$isthebinomialideal$J_G \subset K[X, Y]$whichisgeneratedbythosebinomials$x_iy_j - x_jy_i$forwhich${i, j}$isanedgeof$G$\mathrm{.}TheHilbertseries$H_{K[X, Y]/J_G}(\lambda)$of$K[X, Y]/J_G$isoftheform$H_{K[X, Y]/J_G}(\lambda) = h_{K[X, Y]/J_G}(\lambda)/(1

• \lambda)^d$,where$d = $\mathrm{d}\mathrm{i}\mathrm{m}$ K[X, Y]/J_G$andwhere$h_{K[X, Y]/J_G}(\lambda) = h_0 + h_1\lambda + h_2\lambda^2 + \cdots + h_s\lambda^s$witheach$h_i \in $\mathbb{Z}$$andwith$h_s \neq 0$isthe$h$-polynomialof$K[X, Y]/J_G$\mathrm{.}Itisknownthat,when$K[X, Y]/J_G$isCohen-Macaulay,onehas$$\mathrm{r}\mathrm{e}\mathrm{g}$(K[X, Y]/J_G) = $\mathrm{d}\mathrm{e}\mathrm{g}$ h_{K[X, Y]/J_G}(\lambda)$,where$ $\mathrm{r}\mathrm{e}\mathrm{g}$(K[X, Y]/J_G)$isthe(Castelnuovo-Mumford)regularityof$K[X, Y]/J_G$\mathrm{.}Inthepresentpaper,givenarbitraryintegers$r$and$s$with$2 \leq r \leq s$,afinitesimplegraph$G$forwhich$$\mathrm{r}\mathrm{e}\mathrm{g}$(K[X, Y]/J_G) = r$and$$\mathrm{d}\mathrm{e}\mathrm{g}$ h_{K[X, Y]/J_G}(\lambda) = s$ will be constructed.