Finiteness and exponential growth of full graph p-spectra

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≠2, we identify the nonconstant eigenlines with the barycentres of the coordinate-hyperplane cell decomposition of ℝPⁿ⁻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 (3ⁿ−2ⁿ⁺¹+3)/2 distinct spectral values for every fixed p≠2 and every n≥2. 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≥3 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.

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BibTeX RIS

BibTeX

@misc{colbrook2026finiteness,
  title = {{Finiteness and exponential growth of full graph $p$-spectra}},
  author = {Matthew J. Colbrook},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20291},
  eprint = {2609.20291},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20291},
}

RIS

TY  - UNPB
TI  - Finiteness and exponential growth of full graph p-spectra
AU  - Matthew J. Colbrook
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.20291
UR  - https://arxiv.org/abs/2609.20291
AB  - 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≠2, we identify the nonconstant eigenlines with the barycentres of the coordinate-hyperplane cell decomposition of 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≠2 and every n≥2. 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≥3 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.
ER  -

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