bibkey: reyes2023tokenspectra
authors: M. A. Reyes and C. Dalfó and M. À. Fiol
year: 2024
title: On the spectra and spectral radii of token graphs
doi: 10.1007/s40590-023-00583-3
url: https://arxiv.org/html/2310.16929v1
claim: "Definition 3.1 takes extrema of consecutive Laplacian spectral multiset differences; Conjecture 3.3 proposes monotonicity of their maxima."
strata_touched:
  - D5/S3/Combinatorics/Graph/TokenGraph/NewLayerMaximaRefutation
license: citation-only
triage: anchor

On the spectra and spectral radii of token graphs

The introduction defines F_k(G) on the k-element subsets of V(G). Two configurations are adjacent exactly when their symmetric difference is an edge of G, or equivalently one token moves to an unoccupied vertex along an edge. Section 2.1 fixes simple connected graphs and the ordinary Laplacian L = D − A. Section 3 explicitly uses Laplacian spectra, sets F_0(G) = K_1, and records multiplicity-preserving spectral containment (5).

Definition 3.1 defines α_k and ρ_k as the minimum and maximum of the multiset sp F_k(G) minus sp F_(k−1)(G), for 1 ≤ k ≤ floor(n/2). The following sentence justifies nonemptiness using the strict inequality choose(n,k) > choose(n,k−1). Conjecture 3.3 states:

For any graph G, ρ₁(G) ≤ ρ₂(G) ≤ ⋯ ≤ ρ⌊n/2⌋(G).

Primary source: arXiv:2310.16929v1, Section 3, Definition 3.1 and Conjecture 3.3; versioned PDF. The journal article, Boletín de la Sociedad Matemática Mexicana, volume 30, article 11 (2024), Section 3, retains Definition 3.1 and Conjecture 3.3.

Verified locator

• DOI: https://doi.org/10.1007/s40590-023-00583-3, Boletín de la Sociedad Matemática Mexicana 30, article 11 (2024).
• arXiv HTML: https://arxiv.org/html/2310.16929v1, Section 3, Definition 3.1 and Conjecture 3.3.
• The DOI article is the journal version of the arXiv source; both locators identify the consecutive spectral-multiset definition and the successive-maxima conjecture used here.

The neighboring Conjecture 3.2 concerns minima, and is a different claim. The full-sector spectral radii in equation (7) are different quantities from these new-layer maxima.

Dongsheng Wei, Generators of stability-preserving semigroups, spectral gaps, and classical ground-state computation, arXiv:2609.26802v1 §4.6 and v2 §7, supplies prior star K₁,₃ / two-token C₆ and full-sector spectral interpolation ingredients. The inspected passages distinguish full-sector and new-layer quantities; no novelty of these ingredients or of the spectral method is asserted. The residual-layer conjecture is the target here, rather than the full-radius equality characterization in arXiv:2610.00500 Conjecture 1.1.