{
  "abstract": "Let $Λ_n$ be the set of all eigenvalues of $n\\times n$ standardized Laplacian matrices, and let $S_n\\subseteqΛ_n$ be the polygon introduced in 2005, with upper vertices $λ_0=0,\\dots,λ_{n-1}=1$. It is now known that $Λ_n=S_n$, and that $Λ_n$ is convex, if and only if $n\\le4$. We study how $Λ_n$ leaves $S_n$ near an edge $[λ_m,λ_{m+1}]$, $2(m+1)<n$. First, we prove a switching law for matrices having a boundary point of $Λ_n$ as a simple eigenvalue: an explicit switching function on the arcs, built from the left and right eigenvectors and the outward normal, forces maximal weight where positive and zero weight where negative; intermediate weights occur only where it vanishes. Second, we perturb the weights of the arcs $k\\to k+m+1$ in the circulant digraph (arcs $k\\to k+s$, $1\\le s\\le m$, plus these arcs with weight $τ$) that realizes the point $λ_m+τ(λ_{m+1}-λ_m)$. At second order, the first Fourier harmonic of the perturbation moves the eigenvalue beyond the line of the edge if and only if $τ<τ^*_m=\\sin(πm/n)/(2\\cos(π/n)\\sin(π(m+1)/n))$. For even $n$ and $m$, the point with $τ=τ^*_m$ is the vertex of $S_{n/2}$ lying on this edge, and digraphs with these arcs only at even $k$ give an explicit eigenvalue path that leaves $S_n$ at $λ_m$ and, in all computed cases, ends at this vertex. Third, we give a conjectural, numerically supported picture of $\\partialΛ_5$: beyond each edge of $S_5$ not incident to $0$ or $1$, $Λ_5$ forms a cap bounded by four algebraic arcs, which leaves the edge tangentially at its endpoint adjacent to $0$ or $1$ and rejoins it at $τ^*_1=(3-\\sqrt5)/2$; the rest of the edge lies on $\\partialΛ_5$. In computations for $n\\le8$, a cap returns to its edge smoothly at $τ^*_m$, at a corner at $τ^*_m$, or at a corner beyond $τ^*_m$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02249",
  "author_total": 1,
  "authors": [
    {
      "name": "Pavel Chebotarev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dhz45cpr2pngf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02249",
  "fulltext_url": null,
  "id": "dhz45cpr2pngf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02249",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02249",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dhz45cpr2pngf.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02249",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02249"
    }
  ],
  "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": "How the eigenvalue region of standardized Laplacians leaves the 2005 polygon",
  "title_text": "How the eigenvalue region of standardized Laplacians leaves the 2005 polygon",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/dhz45cpr2pngf",
  "venue": "arXiv",
  "versions": []
}