Regularity and $h$-polynomials of monomial ideals

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form $h_{S/I}(\lambda)/(1 - \lambda)^d$, where $h_{S/I}(\lambda) = h_0 + h_1\lambda + h_2\lambda^2 + \cdots + h_s\lambda^s$ with $h_s \neq 0$ is the $h$-polynomial of $S/I$. It is known that, when $S/I$ is Cohen--Macaulay, one has $\reg(S/I) = \deg h_{S/I}(\lambda)$, where $\reg(S/I)$ is the (Castelnuovo--Mumford) regularity of $S/I$. In the present paper, given arbitrary integers $r$ and $s$ with $r \geq 1$ and $s \geq 1$, a monomial ideal $I$ of $S = K[x_1, \ldots, x_n]$ with $n \gg 0$ for which $\reg(S/I) = r$ and $\deg h_{S/I}(\lambda) = s$ will be constructed. Furthermore, we give a class of edge ideals $I \subset S$ of Cameron--Walker graphs with $\reg(S/I) = \deg h_{S/I}(\lambda)$ for which $S/I$ is not Cohen--Macaulay.