Induced matching numbers of finite graphs and edge ideals

Let $G$ be a finite simple graph on the vertex set $V(G) = {x_1, ldots, x_n}$ and $I(G) subset K[V(G)]$ its edge ideal, where $K[V(G)]$ is the polynomial ring in $x_1, ldots, x_n$ over a field $K$ with each ${rm deg} x_i = 1$ and where $I(G)$ is generated by those squarefree quadratic monomials $x_ix_j$ for which ${x_i, x_j}$ is an edge of $G$. In the present paper, given integers $1 leq a leq r$ and $s geq 1$, the existence of a finite connected simple graph $G = G(a, r, d)$ with ${rm im}(G) = a$, ${rm reg}(R/I(G)) = r$ and ${rm deg} h_{K[V(G)]/I(G)} (lambda) = s$, where ${rm im}(G)$ is the induced matching number of $G$ and where $h_{K[V(G)]/I(G)} (lambda)$ is the $h$-polynomial of $K[V(G)]/I(G)$.