{
  "abstract": "Let $G$ be a finite simple graph and let $I_c(G)$ denote its complementary edge ideal. In this paper, we study the Castelnuovo-Mumford regularity, the depth and Serre depth for $(S_{2})$ of the symbolic powers of $I_c(G)$. We derive explicit formulas for $\\operatorname{reg} I_c(G)^{(k)}$ and $\\operatorname{reg} I_c(G)^{(k)}/I_c(G)^k$ for every $k\\geq1$, in terms of the number and the sizes of the connected components of $G$. As applications, we determine their asymptotic behavior, and completely characterize the symbolic powers of complementary edge ideals having linear resolutions. We then investigate the symbolic depth function $k\\mapsto\\operatorname{depth} S/I_c(G)^{(k)}$. We determine explicitly its values for $k=2,3,4$, and establish bounds for the asymptotic depth. More generally, for an arbitrary squarefree monomial ideal $I$, we compare the local cohomology modules of $S/I^{(k)}$ for different symbolic powers, and derive corresponding inequalities for the depth and the Serre depth. Applying this to complementary edge ideals, we obtain $\\operatorname{depth} S/I_{c}(G)^{(t)}\\geq\\operatorname{depth} S/I_{c}(G)^{(k)}$ for $k\\geq 2t-1$, despite the fact that the symbolic depth function need not be non-increasing. Finally, we characterize complementary edge ideals satisfying Serre's condition $(S_r)$, and determine the Serre depth for $(S_{2})$ of all symbolic powers of complementary edge ideals. In particular, we obtain combinatorial characterizations of Serre's condition $(S_{2})$ and of the Cohen-Macaulayness of $S/I_{c}(G)^{(k)}$ for every $k\\geq 1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23224",
  "author_total": 3,
  "authors": [
    {
      "name": "Antonino Ficarra"
    },
    {
      "name": "Somayeh Moradi"
    },
    {
      "name": "Yuji Muta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4ss2zknq105a7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23224",
  "fulltext_url": null,
  "id": "4ss2zknq105a7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23224",
  "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": "W7222985115",
  "pdf_url": "https://arxiv.org/pdf/2610.23224",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4ss2zknq105a7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23224",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23224"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Regularity, depth and Serre depth of symbolic powers of complementary edge ideals",
  "title_text": "Regularity, depth and Serre depth of symbolic powers of complementary edge ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/4ss2zknq105a7",
  "venue": "arXiv",
  "versions": []
}