{
  "abstract": "Let $G_{n,m}=\\overline{\\C_n^{[m]}}$, where $\\C_n^{[m]}$ denotes the closed $m$th power of the $n$-cycle. We study the graded Betti numbers and homological invariants of the edge ring of $G_{n,m}$ in the range $n\\geq 3m+1$. These graphs form a natural family for the study of edge rings whose regularity can be compared explicitly with the induced matching number. In particular, for $n\\geq4m+1$, the graph $G_{n,m}$ has induced matching number one, whereas its edge ring has regularity two. Our approach is based on a characterization of the homology of the induced subcomplexes of the independence complex $Δ(G_{n,m})$. We introduce a family $\\mathcal{S}_V(k,m)$ of vertex subsets characterized by their successive gaps around the cycle and show that, for $W\\in\\mathcal{S}_V(k,m)$, the induced subcomplex $Δ[W]$ has the homotopy type of $\\mathbb{S}^1$, whereas for $W\\notin\\mathcal{S}_V(k,m)$ all its positive-dimensional reduced homology groups vanish. Combining this characterization with Hochster's formula and an explicit enumeration of $\\mathcal{S}_V(k,m)$, we obtain a closed formula for the graded Betti numbers in the second strand. We further determine the extremal Betti number, regularity, and projective dimension of the edge ring of $G_{n,m}$. Finally, we compute the $f$- and $h$-vectors of the independence complex and use the Hilbert series to determine the graded Betti numbers in the linear strand. The case $m=2$ recovers the corresponding results for complements of squares of cycles obtained in~\\cite{RatherSquare}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17651",
  "author_total": 3,
  "authors": [
    {
      "name": "Shahnawaz Ahmad Rather"
    },
    {
      "name": "S. Pirzada"
    },
    {
      "name": "M. Aijaz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d95bkhtab3ck9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17651",
  "fulltext_url": null,
  "id": "d95bkhtab3ck9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17651",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7213508832",
  "pdf_url": "https://arxiv.org/pdf/2609.17651",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d95bkhtab3ck9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17651",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17651"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Homological invariants of Edge Ideals associated to powers of cycles",
  "title_text": "Homological invariants of Edge Ideals associated to powers of cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/d95bkhtab3ck9",
  "venue": "arXiv",
  "versions": []
}