{
  "abstract": "For some positive integer $m$, a real polynomial $P(x)=\\sum\\limits_{k=0}^ma_kx^k$ with $a_k\\geqslant 0$ is called {log-concave} (resp. ultra log-concave) if $a_k^2\\geqslant a_{k-1}a_{k+1}$ (resp. $a_k^2\\geqslant \\left(1+\\frac{1}{k}\\right)\\left(1+\\frac{1}{m-k}\\right)\\cdot$ $a_{k-1}a_{k+1}$) for all $1\\leqslant k\\leqslant m-1$. If $P(x)$ has only real roots, then it is called {real-rooted}. It is well-known that the conditions of log-concavity, ultra log-concavity and real-rootedness are ever-stronger. %A famous theorem due to Newton states that if a real polynomial with non-negative coefficients is real-rooted, then it is log-concave. For a graph $G$, a dependent set is a set of vertices which is not independent, i.e., the set of vertices whose induced subgraph contains at least one edge. The dependence polynomial of $G$ is defined as $D(G, x):=\\sum\\limits_{k\\geqslant 0}d_k(G)x^k$, where $d_k(G)$ is the number of dependent sets of size $k$ in $G$. Horrocks proved that $D(G, x)$ is log-concave for every graph $G$ [J. Combin. Theory, Ser. B, 84 (2002) 180--185]. In the present paper, we prove that, for a graph $G$, $D(G, x)$ is ultra log-concave if $G$ is $(K_2\\cup 2K_1)$-free or contains an independent set of size $|V(G)|-2$, and give the characterization of graphs whose dependence polynomials are real-rooted. Finally, we focus more attention to the problems of log-concavity about independence systems and pose several conjectures closely related the famous Mason's Conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "Yan-Ting Xie"
    },
    {
      "name": "Shou-Jun Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4764rshmn6ar3.bib",
  "cited_by": 0,
  "doi": "10.37236/13310",
  "fulltext_url": null,
  "id": "4764rshmn6ar3",
  "issns": [
    "1077-8926"
  ],
  "landing_url": "https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p9",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W4402510033",
  "pdf_url": "https://www.combinatorics.org/ojs/index.php/eljc/article/download/v33i4p9/pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": "The Electronic Journal of Combinatorics",
  "ris": "https://research.draconitic.com/paper/4764rshmn6ar3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:24Z",
      "native_id": "10.37236/13310",
      "source": "crossref",
      "url": "https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p9"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2607|Advanced Combinatorial Mathematics",
      "subfield": "combinatorics"
    }
  ],
  "title": "Ultra Log-Concavity and Real-Rootedness of Dependence Polynomials",
  "title_text": "Ultra Log-Concavity and Real-Rootedness of Dependence Polynomials",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:24Z",
  "url": "https://research.draconitic.com/paper/4764rshmn6ar3",
  "venue": "The Electronic Journal of Combinatorics",
  "versions": []
}