{
  "abstract": "In this paper, we introduce and investigate the $\\textbf{k}$-neighborhood ideal of a graph, a natural generalization of the closed neighborhood ideal. Let $G$ be a simple graph on the vertex set $[n]$, and let $S=K[x_1,\\dots,x_n]$ be the polynomial ring over a field $K$. For a vector $\\textbf{k}=(k_1,\\ldots,k_n)\\in \\mathbb{N}^n$ satisfying $1\\leq k_i\\leq \\textrm{deg}_G(i)+1$ for all $i$, the $\\textbf{k}$-neighborhood ideal of $G$ is defined as the squarefree monomial ideal $$\\textrm{NI}_{\\textbf{k}}(G)=\\sum_{i=1}^n\\, (\\textbf{x}_W:\\, W\\subseteq N_G[i],\\, |W|=k_i)$$ of $S$, where $\\textbf{x}_W=\\prod_{i\\in W} x_i$. We study homological invariants and properties of $\\textrm{NI}_{\\textbf{k}}(G)$ focusing on its Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness. Special attention is devoted to the case where the vector ${\\textbf{k}}$ is the degree-vector of the graph, i.e., $k_i=\\textrm{deg}_G(i)$ for all vertices $i$, and to the case where $\\textrm{NI}_{\\textbf{k}}(G)$ coincides with the edge ideal of a graph. In these settings, we provide combinatorial characterizations and bounds for the regularity and projective dimension of $\\textrm{NI}_{\\textbf{k}}(G)$ for several classes of graphs, and further investigate the Cohen-Macaulay property of these ideals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.08159",
  "author_total": 2,
  "authors": [
    {
      "name": "Somayeh Moradi"
    },
    {
      "name": "Leila Sharifan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vnhy541vncc0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.08159",
  "fulltext_url": null,
  "id": "3vnhy541vncc0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.08159",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7164016634",
  "pdf_url": "https://arxiv.org/pdf/2606.08159",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vnhy541vncc0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2606.08159",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.08159"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "$\\textbf{k}$-neighborhood ideals of graphs",
  "title_text": "k-neighborhood ideals of graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/3vnhy541vncc0",
  "venue": "arXiv",
  "versions": []
}