bibkey: shapiro2026spectrum authors: Boris Shapiro year: 2026 title: The (n−2,2)-Spectrum of a Graph doi: null url: https://arxiv.org/abs/2605.17501v2 claim: The unweighted fourth-moment inversion question asks whether Laplacian and cubic data recover the four-edge support-forest counts. strata_touched:
- D5/S3/Combinatorics/Graph/SupportForestMoments/ShapiroQuarticInversionRefutation license: citation-only triage: anchor
Verified locator
DOI: null. Crossref’s title query returned no matching work. Source: https://arxiv.org/abs/2605.17501v2
Section 6, page 4, equations (6.1) and (6.2), defines the character and edge-word expansion of the trace moment. Section 6.1, page 5, defines support-subgraph counts:
For a finite simple graph H without isolated vertices, let N_H(G) denote the number of edge subsets S ⊆ E(G) for which the graph with edge set S and vertex set formed by the endpoints of S is isomorphic to H. No inducedness condition is imposed on the ambient graph G.
The cycle-count convention in Section 6, page 4, is:
If σ ∈ Sₙ, let c₁(σ) be the number of fixed points of σ and let c₂(σ) be the number of two-cycles in its cycle decomposition.
The edge transposition is defined there by:
Let τ_e = (ij) denote the transposition corresponding to an edge e = ij.
Section 9, page 11, discusses degree-by-degree inversion after adjoining Laplacian spectral data and distinguishes recovery from all moments. Section 12, page 13, Outlook item 1 states:
Compute an explicit closed formula for the unweighted fourth moment and invert it, modulo Laplacian and cubic data, on the finite list of four-edge support forests. The degree-three inversion is complete by the cubic inversion theorem above.
Scope
The inversion implication compares trees of the same order, with equal Laplacian characteristic polynomials, equal support counts through three edges, and equal moments of orders two, three and four. The counterexample has unequal five-vertex path counts. This answers the inversion part of item 1; it does not preclude an explicit fourth-moment formula. The all-moment separation statements remain distinct questions.