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$I(G) \subset K[V(G)]$itsedgeideal,where$K[V(G)]$ is the polynomial ring in $x_1,\dots ,x_n$ over a field $K$ with each ${\rm deg} x_i = 1$andwhere$I(G)$isgeneratedbythosesquarefreequadraticmonomials$x_ix_j$forwhich${x_i, x_j}$isanedgeof$G$. In the present paper, given integers $1\le a\le r$ and $s\ge 1$, the existence of a finite connected simple graph $G=G(a,r,d)$ with $\mathrm{i}\mathrm{m}(G)=a$, ${\rm reg}(R/I(G)) = r$and${\rm deg} h_{K[V(G)]/I(G)} (\lambda) = s$,where${\rm im}(G)$istheinducedmatchingnumberof$G$andwhere$h_{K[V(G)]/I(G)} (\lambda)$isthe$h$-polynomialof$K[V(G)]/I(G)$.