Regularity and \(h\)-polynomials of edge ideals (Q668077)

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