Regularity and $h$-polynomials of monomial ideals

Let $S=K[x_1,\dots ,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\mathrm{deg}{x}_i=1$ and $I\subset S$ a homogeneous ideal of $S$ with $\dim S\mathrm{/}I=d$. The Hilbert series of $S\mathrm{/}I$ is of the form $h_{S\mathrm{/}I}(\lambda )\mathrm{/}(1-\lambda {)}^d$, where $h_{S/I}(\lambda) = h_0 + h_1\lambda

• h_2\lambda^2 + \cdots + h_s\lambda^s$with$h_s \neq 0$isthe$h$-polynomial of $S\mathrm{/}I$. It is known that, when $S\mathrm{/}I$ is Cohen–Macaulay, one has $\reg(S/I) = \deg h_{S/I}(\lambda)$,where$\reg(S/I)$ is the (Castelnuovo–Mumford) regularity of $S\mathrm{/}I$. In the present paper, given arbitrary integers $r$ and $s$ with $r\ge 1$ and $s\ge 1$, a monomial ideal $I$ of $S = K[x_1, \ldots, x_n]$with$n \gg 0$forwhich$\reg(S/I) = r$and$\deg h_{S/I}(\lambda) = s$willbeconstructed\mathrm{.}Furthermore,wegiveaclassofedgeideals$I \subset S$ofCameron--Walkergraphswith$\reg(S/I) = \deg h_{S/I}(\lambda)$forwhich$S/I$ is not Cohen–Macaulay.