{
  "abstract": "Let $Λ_n$ be the set of all eigenvalues of standardized Laplacian matrices of order $n$, equivalently, of stochastic $n\\times n$ matrices with off-diagonal entries at most $\\frac1n$. In 2005, a polygon $S_n\\subseteqΛ_n$ was constructed, and it was conjectured that $Λ_n=S_n$; this holds for $n\\le3$. We show that the conjecture fails for every $n\\ge5$. The vertices of $S_n$ are eigenvalues of circulant Laplacians built from powers of the directed $n$-cycle. To first order, no perturbation of these circulants moves such an eigenvalue out of $S_n$; however, adding chords with nonconstant weights moves it along an edge of $S_n$ to first order and, with an explicit coefficient, beyond the supporting line of this edge to second order. Consequently, every edge of $S_n$ not incident to $0$ or $1$ contains interior points of $Λ_n$, except possibly, for even $n$, the two horizontal edges, whereas the four edges incident to $0$ or $1$ lie on $\\partialΛ_n$. For $n=4$, whose only other edges are horizontal, the question was left open in the previous versions; it has since been settled, $Λ_4=S_4$, by a computer-assisted proof of L. Marshall, after an earlier claim by D.Z. Liu, who also proved that $Λ_n$ is not convex for $n\\ge5$. We certify an explicit weighted counterexample for $n=5$, lying only $1.5\\cdot10^{-3}$ outside $S_5$, and an unweighted one for $n=7$: the directed $6$-cycle with a duplicated vertex. Its eigenvalue outside $S_7$ is also an eigenvalue of the two-chord digraph that ends the above perturbation; for the analogous digraphs of order $n$, the excess is asymptotic to $\\frac{π^2}{2n^4}$. We also observe that $S_n$ is a zonotope, describe $Λ_n$ in terms of eigenvectors, and prove that $S_m\\subseteq S_n$ whenever $m\\mid n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14100",
  "author_total": 1,
  "authors": [
    {
      "name": "Pavel Chebotarev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6cpc8agq0fvvn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14100",
  "fulltext_url": null,
  "id": "6cpc8agq0fvvn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14100",
  "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"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7221812429",
  "pdf_url": "https://arxiv.org/pdf/2610.14100",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6cpc8agq0fvvn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.14100",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14100"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Digraph Laplacian spectra: The 2005 polygon conjecture fails for all orders above four",
  "title_text": "Digraph Laplacian spectra: The 2005 polygon conjecture fails for all orders above four",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/6cpc8agq0fvvn",
  "venue": "arXiv",
  "versions": []
}