{"paper":{"title":"The (n-2,2)-Spectrum of a Graph","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A weighted trace polynomial from the (n-2,2) representation reconstructs every tree from its second moment except for one exceptional n.","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Boris Shapiro","submitted_at":"2026-05-17T15:22:15Z","abstract_excerpt":"We study a representation-theoretic refinement of the ordinary Laplacian spectrum of a graph. Given a graph $G$ on $n$ vertices, one may associate to it the element \\[ X_G=\\sum_{ij\\in E(G)} (ij)\\in \\C[S_n]. \\] The action of $X_G$ in irreducible representations of $S_n$ produces spectral invariants of graphs. The standard representation $(n-1,1)$ recovers the ordinary graph Laplacian spectrum, up to the elementary affine change $X_G=mI-L_G$, where $m=|E(G)|$. The next component, $(n-2,2)$, gives the first representation-theoretic correction. We give an explicit edge-space model for this compone"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The weighted trace polynomial already reconstructs every tree from the second moment, except for a single exceptional value of n where the fourth moment suffices.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The explicit edge-space model for the (n-2,2) component induces an operator whose trace moments are exactly the stated linear combinations of support-forest counts, with no hidden dependencies on the choice of basis or on graph-specific normalizations.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Defines the (n-2,2) component of the S_n-representation of the graph element X_G, derives its edge-space model and trace moments in terms of support-forest counts, and proves that a weighted trace polynomial reconstructs all trees from the second moment except for one exceptional n.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A weighted trace polynomial from the (n-2,2) representation reconstructs every tree from its second moment except for one exceptional n.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"18058f77a062ba4f9ed70f2452ee97beddc7239608408af2faf704702f8d24e2"},"source":{"id":"2605.17501","kind":"arxiv","version":1},"verdict":{"id":"6c6b1e13-69c6-4e6d-b13d-c1ee00799300","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:36:23.171624Z","strongest_claim":"The weighted trace polynomial already reconstructs every tree from the second moment, except for a single exceptional value of n where the fourth moment suffices.","one_line_summary":"Defines the (n-2,2) component of the S_n-representation of the graph element X_G, derives its edge-space model and trace moments in terms of support-forest counts, and proves that a weighted trace polynomial reconstructs all trees from the second moment except for one exceptional n.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The explicit edge-space model for the (n-2,2) component induces an operator whose trace moments are exactly the stated linear combinations of support-forest counts, with no hidden dependencies on the choice of basis or on graph-specific normalizations.","pith_extraction_headline":"A weighted trace polynomial from the (n-2,2) representation reconstructs every tree from its second moment except for one exceptional n."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17501/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.522299Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:41:16.933953Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:41:57.666693Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.636517Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"1a2c0c5e20b1959fedfb4a5e4a852ac950aab1515345761367d648745d8f8e9b"},"references":{"count":10,"sample":[{"doi":"","year":1988,"title":"Diaconis,Group Representations in Probability and Statistics, Institute of Mathematical Statistics Lecture Notes– Monograph Series, vol","work_id":"0a906be5-84f0-46aa-b259-95cb85992c67","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2001,"title":"C. Godsil and G. Royle,Algebraic Graph Theory, Graduate Texts in Mathematics, vol. 207, Springer, New York, 2001","work_id":"fb86b010-e6c5-47ff-ad19-a99725e9f20c","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1982,"title":"C. D. Godsil and B. D. McKay,Constructing cospectral graphs, Aequationes Math.25(1982), 257–268","work_id":"74ef949a-06a8-4634-9df5-b9da1f004acd","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"W. H. Haemers and E. Spence,Enumeration of cospectral graphs, European J. Combin.25(2004), no. 2, 199–211","work_id":"a96a22aa-8699-4afa-bae4-b1301f538f0c","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"B. D. McKay and A. Piperno,Practical graph isomorphism, II, J. 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