On homological invariants and Cohen-Macaulayness of closed neighborhood ideals

Abstract: Let $G$ be a finite simple graph and $NI(G)$ be the closed neighborhood ideal of $G$ in the polynomial ring $S=K[V(G)]$. In this paper, we study the Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness of this ideal. For any chordal graph $G$, we show that $\text{reg}(S/NI(G))=τ(G)$, where $τ(G)$ denotes the vertex cover number of $G$. This generalizes the corresponding result for trees shown in \cite{CJRS}, as in trees $τ(G)$ is the same as the matching number of $G$. When $G$ is a bipartite graph or a very well-covered graph, we notice that $\text{reg}(S/NI(G))\geq τ(G)$ and that this inequality can be strict in general. Moreover, we describe the projective dimension of $S/NI(G)$ for some families of graphs. Finally, we give a characterization of very well-covered graphs $G$ for which the ring $S/NI(G)$ is Cohen-Macaulay.