{
  "abstract": "For a simple graph $G$ with $n$ vertices, write its chromatic polynomial in the rising factorial basis as $$ χ_G(x)=\\sum_{i=0}^{n}(-1)^{n-i}c_i(G)\\langle x\\rangle_i,$$ where $ \\langle x\\rangle_i=x(x+1)\\cdots(x+i-1).$ The associated $τ$-polynomial $$ τ_G(x)=\\sum_{i=0}^{n}c_i(G)x^i $$ was defined and systematically investigated by Brenti in 1992. In this paper, we prove that if the $τ$-polynomials of two vertex-disjoint simple graphs $G$ and $H$ have only real zeros, then the $τ$-polynomial of their join $G\\vee H$ has only real zeros. This settles a conjecture posed by Brenti, Royle and Wagner since 1994.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07457",
  "author_total": 4,
  "authors": [
    {
      "name": "Mingyang Kang"
    },
    {
      "name": "Zhixin Liu"
    },
    {
      "name": "Sophie C. C. Sun"
    },
    {
      "name": "Philip B. Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6h3j2jjvgse1x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07457",
  "fulltext_url": null,
  "id": "6h3j2jjvgse1x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07457",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212069137",
  "pdf_url": "https://arxiv.org/pdf/2609.07457",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6h3j2jjvgse1x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07457",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07457"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Real-rootedness of the $τ$-polynomial under graph joins",
  "title_text": "Real-rootedness of the τ-polynomial under graph joins",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/6h3j2jjvgse1x",
  "venue": "arXiv",
  "versions": []
}