Regularity and $h$-polynomials of edge ideals

TL;DR

This paper constructs specific edge ideals with prescribed regularity and $h$-polynomial degree, and establishes an upper bound relating these invariants to the number of vertices in the graph.

Contribution

It demonstrates the existence of edge ideals with arbitrary regularity and $h$-polynomial degree, and proves a new bound connecting these parameters to the graph's size.

Findings

Existence of edge ideals with prescribed regularity and $h$-polynomial degree.

Upper bound of regularity plus $h$-polynomial degree by the number of vertices.

Insight into the relationship between algebraic invariants and graph size.

Abstract

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