Castelnuovo-Mumford regularity of the closed neighborhood ideal of a graph

Abstract

Let $G$ be a finite simple graph and let $NI(G)$ denote the closed neighborhood ideal of $G$ in a polynomial ring $R$. We show that if $G$ is a forest, then the Castelnuovo-Mumford regularity of $R/NI(G)$ is the same as the matching number of $G$, thus proving a conjecture of Sharifan and Moradi in the affirmative. We also show that the matching number of $G$ provides a lower bound for the Castelnuovo-Mumford regularity of $R/NI(G)$ for any $G$. Furthermore, we prove that, if $G$ contains a simplicial vertex, then $NI(G)$ admits a Betti splitting, and consequently, we show that the projective dimension of $R/NI(G)$ is also bounded below by the matching number of $G$, if $G$ is a forest or a unicyclic graph.