{
  "abstract": "We prove that the full spectrum of the graph $p$-Laplacian is finite for every finite graph and every real $p>1$. This resolves the problem of finiteness of the full graph spectrum posed by Amghibech (2003). More generally, for signed weighted graph $p$-Schrödinger operators with arbitrary real potentials, we obtain bounds in terms of the numbers of vertices and positive-weight edges, uniform in all coefficients and in $p$. On $n$ vertices with $m$ positive-weight edges, the logarithms of the spectral cardinality and of the total number of connected components of the normalised eigenvector sets are $O((n+m)^2)$; the sum over the full spectrum of their rational Betti numbers is at most $\\exp(C(n+m)^2)$ for an absolute constant $C$. These topological bounds extend to homogeneous eigenproblems built from arbitrary finite families of linear forms, including generalised $p$-eigenvalue problems for matrix pairs. For the uniformly weighted $K_n$ and $p\\ne2$, we identify the nonconstant eigenlines with the barycentres of the coordinate-hyperplane cell decomposition of $\\mathbb RP^{n-2}$ and determine their local Morse data on either side of $p=2$. A renormalised logarithmic limit describes the transition at $p=2$. Positive integer weights then separate these critical values, giving at least $(3^n-2^{n+1}+3)/2$ distinct spectral values for every fixed $p\\ne2$ and every $n\\geq2$. By contrast, the maximal spectral cardinality at $p=2$ is $n$; at $p=4$, the maximal spectral cardinalities in the unsigned and general classes both have exponential growth rate exactly $3$. For $n\\geq3$ the same examples resolve Amghibech's extremal question. The proof combines o-minimal and Pfaffian geometry with projective $L^p$-duality, Morse theory, critical groups, and tensor eigenvalue bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20291",
  "author_total": 1,
  "authors": [
    {
      "name": "Matthew J. Colbrook"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8eqj6skkkk9xk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20291",
  "fulltext_url": null,
  "id": "8eqj6skkkk9xk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20291",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.20291",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8eqj6skkkk9xk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:19Z",
      "native_id": "2609.20291",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20291"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Finiteness and exponential growth of full graph $p$-spectra",
  "title_text": "Finiteness and exponential growth of full graph p-spectra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/8eqj6skkkk9xk",
  "venue": "arXiv",
  "versions": []
}