  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 if $G$ contains a simplicial vertex, then
$NI(G)$ admits a Betti splitting, and consequently, the matching number of $G$
provides a lower bound for the Castelnuovo-Mumford regularity of $R/NI(G)$ when
$G$ is a chordal graph, unicyclic graph, complete bipartite graph, or the wheel
graph. For forests and unicyclic graphs, we show that the projective dimension
of $R/NI(G)$ is also bounded below by the matching number of $G$. Moreover, we
investigate the relationship between the regularity and the matching number for
two graph operations, namely, the join and the corona product of two graphs.
