Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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FALSE: the matrix-tree theorem works only for one distinguished cofactor

Statement

False claim. The matrix-tree theorem computes the spanning-tree count from only one special cofactor of the Laplacian; deleting a different row and column can change the answer.

Facts & Assumptions

Given: A finite simple graph G.

[L1]

Every principal cofactor of the Laplacian equals τ(G) (Kirchhoff's matrix-tree theorem).

Refutation

technique · direct
1.1

By [L1], for every vertex index i the principal cofactor obtained by deleting row i and column i has determinant τ(G). So the value does not depend on a distinguished choice of index.

L1
2.1

This is exactly the negation of the false claim, so the claim is refuted.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources