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Let $G$ be a finite simple graph, and $NI(G)$ denote the closed neighborhoodideal of $G$ in a polynomial ring $R$. We show that if $G$ is a forest, thenthe Castelnuovo-Mumford regularity of $R/NI(G)$ is the same as the matchingnumber of $G$, thus proving a conjecture of Sharifan and Moradi in theaffirmative. We also show that the matching number of $G$ provides a lowerbound for the Castelnuovo-Mumford regularity of $R/NI(G)$ when $G$ is a chordalgraph, complement of a tree, complete bipartite graph, cycle graph, or a wheelgraph. Moreover, we investigate the relationship between these two invariantsfor two graph operations, namely, the join and the corona product of graphs.

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