{
  "abstract": "In this paper, we resolve a 30-year-old conjecture of Spielman and Teng concerning the performance of the spectral partitioning method on graphs embeddable on an orientable surface of genus $g\\ge 1$. In particular, for such a graph $G$ with $n$ vertices and maximum degree $Δ$, we show that the second-smallest eigenvalue of its Laplacian matrix satisfies $λ_2(L_G)\\lesssimΔ\\frac g n$. We also obtain an improved eigenvalue bound for $K_h$-minor-free graphs of $λ_2(L_G)\\lesssimΔ\\frac{h^2(\\log h)^2}n$. In fact, our results directly prove much stronger results for reweighted eigenvalues, including higher reweighted eigenvalues. As a consequence, we obtain bounds not just on Laplacian eigenvalues, but also on normalized Laplacian eigenvalues and Steklov eigenvalues. Our results for genus-$g$ graphs are optimal for all of these kinds of eigenvalues, while our results for $K_h$-minor-free graphs are optimal up to $\\log(h)$ factors. Our techniques for genus-$g$ graphs bootstrap bounded-degree bounds of normalized eigenvalues for entire classes to bounds for reweighted eigenvalues for the same classes without the bounded-degree limitation, while our techniques for $K_h$-minor-free graphs generalize an argument of Korhonen and Lokshtanov, making use of the Lovász local lemma.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27179",
  "author_total": 2,
  "authors": [
    {
      "name": "Benedikt Kolbe"
    },
    {
      "name": "Jack Spalding-Jamieson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2dbgwa3963dhg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27179",
  "fulltext_url": null,
  "id": "2dbgwa3963dhg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27179",
  "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"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7204671255",
  "pdf_url": "https://arxiv.org/pdf/2608.27179",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2dbgwa3963dhg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2608.27179",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27179"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On Eigenvalue Bounds for Bounded Genus Graphs and Minor-Free Graphs",
  "title_text": "On Eigenvalue Bounds for Bounded Genus Graphs and Minor-Free Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/2dbgwa3963dhg",
  "venue": "arXiv",
  "versions": []
}