Induced matching numbers of finite graphs and edge ideals

Abstract

Let $G$ be a finite simple graph on the vertex set $V(G)=\{x_1,\dots ,x_n\}$ and $I(G)\subset K[V(G)]$ its edge ideal, where $K[V(G)]$ is the polynomial ring in $x_1,\dots ,x_n$ over a field $K$ with each $\mathrm{deg}{x}_i=1$ and where $I(G)$ is generated by those squarefree quadratic monomials $x_i{x}_j$ for which $\{x_i,x_j\}$ is an edge of $G$. 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 $\mathrm{im}(G)=a$, $\mathrm{reg}(R/I(G))=r$ and $\mathrm{deg}{h}_{K[V(G)]/I(G)}(\lambda )=s$, where $\mathrm{im}(G)$ is the induced matching number of $G$ and where $h_{K[V(G)]/I(G)}(\lambda )$ is the $h$-polynomial of $K[V(G)]/I(G)$.

11 pages