On homological invariants and Cohen-Macaulayness of closed neighborhood ideals
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912897570963456 |
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| author | Moradi, Somayeh Sharifan, Leila |
| author_facet | Moradi, Somayeh Sharifan, Leila |
| contents | 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 [3], 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07910 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On homological invariants and Cohen-Macaulayness of closed neighborhood ideals Moradi, Somayeh Sharifan, Leila Commutative Algebra Combinatorics Primary 13D02, 05E40, Secondary 13C05, 13A02 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 [3], 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. |
| title | On homological invariants and Cohen-Macaulayness of closed neighborhood ideals |
| topic | Commutative Algebra Combinatorics Primary 13D02, 05E40, Secondary 13C05, 13A02 |
| url | https://arxiv.org/abs/2602.07910 |