{
  "abstract": "Bilu and Linial conjectured that every finite $d$-regular graph with $d \\geq 2$ admits an edge signing $σ$ whose signed adjacency matrix $A_σ$ has spectral radius at most $2\\sqrt{d-1}$. We prove that every signing meeting the mixed-root condition $r_{A_σ}\\leq\\sqrt{2(d-1)}$ satisfies \\[ ρ(A_σ) < \\frac{3+\\sqrt5}{2}\\sqrt{d-1}, \\] where $r_{A_σ}$ is the largest root of the mixed determinantal polynomial $χ[A_σ,-A_σ]$. The interlacing theorem of Ravichandran and Srivastava guarantees a signing satisfying the mixed-root condition, so our result improves the coefficient $2\\sqrt2$ in their two-sided spectral bound. In the proof, we construct a positive matrix-valued probability measure supported on the roots of $χ[A_σ,-A_σ]$. The second moment gives a simple matrix inequality $A_σ^2 + dI \\preceq 4r_{A_σ}^2I$, which yields a preliminary coefficient $\\sqrt{7}$. Estimates for the fourth moment use information about short walks to obtain the coefficient $(3+\\sqrt{5})/2$. For triangle-free graphs, the coefficient improves to $\\sqrt6$, and for graphs with girth at least five, it improves to $\\sqrt{35/6}$. As a result of independent interest, we extend the construction to $χ[A_1,\\ldots,A_k]$ for Hermitian matrices $A_1,\\ldots,A_k$ with zero diagonal, and compute the first two moments explicitly. Finally, an explicit signing of $K_8$ shows that the mixed-root condition alone cannot guarantee a coefficient below $(4+\\sqrt5)/\\sqrt6$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15715",
  "author_total": 2,
  "authors": [
    {
      "name": "Fangfang Lin"
    },
    {
      "name": "Hong Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ddn4ybxmfvxwj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15715",
  "fulltext_url": null,
  "id": "ddn4ybxmfvxwj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15715",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7213291327",
  "pdf_url": "https://arxiv.org/pdf/2609.15715",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ddn4ybxmfvxwj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.15715",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15715"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Improved Bounds for the Bilu-Linial Conjecture via Spectral Recovery from Mixed Determinantal Polynomials",
  "title_text": "Improved Bounds for the Bilu-Linial Conjecture via Spectral Recovery from Mixed Determinantal Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/ddn4ybxmfvxwj",
  "venue": "arXiv",
  "versions": []
}