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

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) 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.