{
  "abstract": "We determine the closure of the connected domination roots. The main tool is a substitution formula for the lexicographic product with a complete graph, $D_c(G[K_n],x)=D_c(G,(x+1)^n-1)$, proved in Theorem~\\ref{thm:cd}. Combined with two explicit families of seed roots---the real roots of the cycles $C_n$ and the real roots of the joins $C_m\\vee C_n$ lying in $(-1,0)$---this formula gives the two main results: the closure of the real connected domination roots is $(-\\infty,0]$, and the closure of all connected domination roots is the whole complex plane. These are the connected domination analogues of the root-density theorems of Brown and Tufts and of Brown and Beaton for the ordinary domination polynomial.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06569",
  "author_total": 2,
  "authors": [
    {
      "name": "Pingping Shao"
    },
    {
      "name": "Chengye Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bn3erqy0637vn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06569",
  "fulltext_url": null,
  "id": "bn3erqy0637vn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06569",
  "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",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7212026642",
  "pdf_url": "https://arxiv.org/pdf/2609.06569",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bn3erqy0637vn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.06569",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06569"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the Roots of Connected Domination Polynomials",
  "title_text": "On the Roots of Connected Domination Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/bn3erqy0637vn",
  "venue": "arXiv",
  "versions": []
}