Regularity and $h$-polynomials of edge ideals

Abstract

For any two integers $d,r\ge 1$, we show that there exists an edge ideal $I(G)$ such that the $\mathrm{reg}\left(R/I(G)\right)$, the Castelnuovo-Mumford regularity of $R/I(G)$, is $r$, and $\mathrm{deg}(h_{R/I(G)}(t))$, the degree of the $h$-polynomial of $R/I(G)$, is $d$. Additionally, if $G$ is a graph on $n$ vertices, we show that $\mathrm{reg}\left(R/I(G)\right)+\mathrm{deg}(h_{R/I(G)}(t))\le n$.

10 pages; comments welcomed