Regularity and $h$-Polynomials of Edge Ideals Academic Article

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• Overview
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• Research
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• Identity
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• For any two integers $d,r \geqslant 1$, we show that there exists an edge ideal $I(G)$ such that ${\rm reg}\left(R/I(G)\right)$, 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 ${\rm reg}\left(R/I(G)\right) + \deg h_{R/I(G)}(t) \leqslant n$.