{
  "abstract": "Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the \\emph{partial Petrial polynomial} of a ribbon graph $G$, denoted by $^{\\partial}{\\varepsilon^{\\times}_G}(z)$. For a prime bouquet $B_n$, Yan and Li [Discrete Appl. Math., 375 (2025): 281-289] determined $^{\\partial}{\\varepsilon^{\\times}_{B_n}}(z)$ when the intersection graph $I(B_n)$ is either the complete graph or a path, and provided an equivalent condition under which the lowest degree of the nonzero coefficient in $^{\\partial}{\\varepsilon^{\\times}_{B_n}}(z)$ is $1$. In this paper, we determine $^{\\partial}{\\varepsilon^{\\times}_{B_n}}(z)$ when the intersection graph is a cycle. Moreover, we present a complete characterization of the prime bouquets whose lowest nonzero term in $^{\\partial}{\\varepsilon^{\\times}_{B_n}}(z)$ is of degree $2$ and determine the partial Petrial polynomial for the prime bouquets. As corollaries, we determine $^{\\partial}{\\varepsilon^{\\times}_{B_n}}(z)$ when $I(B_n)$ is the complete bipartite and tripartite graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07570",
  "author_total": 4,
  "authors": [
    {
      "name": "Xiaoxiang Yu"
    },
    {
      "name": "Rong-Xia Hao"
    },
    {
      "name": "Jianbing Liu"
    },
    {
      "name": "Zhiguo Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fxsgerfbwfyxm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07570",
  "fulltext_url": null,
  "id": "fxsgerfbwfyxm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07570",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212055868",
  "pdf_url": "https://arxiv.org/pdf/2609.07570",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fxsgerfbwfyxm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07570",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07570"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Partial Petrial Polynomials of Bouquets",
  "title_text": "Partial Petrial Polynomials of Bouquets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fxsgerfbwfyxm",
  "venue": "arXiv",
  "versions": []
}