{
  "abstract": "For a simple graph $G$, the partial Petrial polynomial $P_G^{\\times}(z)$ enumerates all diagonal perturbations of its adjacency matrix over $GF(2)$ by rank. We derive a rooted attachment identity and apply it to the tree $R_m$ obtained by joining a new vertex to the middle vertex of each of $m$ disjoint copies of the five-vertex path. We prove that the nonzero coefficient sequence of $P_{R_m}^{\\times}(z)$ is strictly unimodal for every $m\\ge1$, but is log-concave if and only if $1\\le m\\le3$. Thus the $21$-vertex tree $R_4$ provides an explicit strictly unimodal, non-log-concave example. For $m\\ge5$, the unique mode occurs at degree $\\lfloor(149m+85)/32\\rfloor$. More generally, for any fixed path rooted at an internal vertex, joining a new vertex to the roots of sufficiently many disjoint copies produces a tree whose partial Petrial polynomial is not log-concave.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07503",
  "author_total": 2,
  "authors": [
    {
      "name": "Ruiqing Feng"
    },
    {
      "name": "Qi Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07d80ne2zhc0x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07503",
  "fulltext_url": null,
  "id": "07d80ne2zhc0x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07503",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07503",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07d80ne2zhc0x.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07503",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07503"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Strict unimodality without log-concavity for partial Petrial polynomials",
  "title_text": "Strict unimodality without log-concavity for partial Petrial polynomials",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:21Z",
  "url": "https://research.draconitic.com/paper/07d80ne2zhc0x",
  "venue": "arXiv",
  "versions": []
}