Pith Number

pith:VASC5TRI

pith:2026:VASC5TRIUFAML4YLTVRJOYTZWL

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The (n-2,2)-Spectrum of a Graph

Boris Shapiro

A weighted trace polynomial from the (n-2,2) representation reconstructs every tree from its second moment except for one exceptional n.

arxiv:2605.17501 v1 · 2026-05-17 · math.CO · math.GR

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

10 extracted · 10 resolved · 0 Pith anchors

[1] Diaconis,Group Representations in Probability and Statistics, Institute of Mathematical Statistics Lecture Notes– Monograph Series, vol 1988

[2] C. Godsil and G. Royle,Algebraic Graph Theory, Graduate Texts in Mathematics, vol. 207, Springer, New York, 2001 2001

[3] C. D. Godsil and B. D. McKay,Constructing cospectral graphs, Aequationes Math.25(1982), 257–268 1982

[4] W. H. Haemers and E. Spence,Enumeration of cospectral graphs, European J. Combin.25(2004), no. 2, 199–211 2004

[5] B. D. McKay and A. Piperno,Practical graph isomorphism, II, J. Symbolic Comput.60(2014), 94–112 2014

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:04:42.470854Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

a8242ece28a140c5f30b9d62976279b2fc294d532d92b108d9a042be57b99fc6

Aliases

arxiv: 2605.17501 · arxiv_version: 2605.17501v1 · doi: 10.48550/arxiv.2605.17501 · pith_short_12: VASC5TRIUFAM · pith_short_16: VASC5TRIUFAML4YL · pith_short_8: VASC5TRI

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/VASC5TRIUFAML4YLTVRJOYTZWL \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: a8242ece28a140c5f30b9d62976279b2fc294d532d92b108d9a042be57b99fc6

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "40d82b5368e9e9d319e5220e4ff85e57df1823532c3e396d1a9cd11a52a63581",
    "cross_cats_sorted": [
      "math.GR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-17T15:22:15Z",
    "title_canon_sha256": "28ec32a24b47e1accb5f6240bb6b6e008a1e708be31b61e2376f43778d1285e9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17501",
    "kind": "arxiv",
    "version": 1
  }
}