Alphabeta MathHow statement and proof provenance work
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.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
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 .
Every principal cofactor of the Laplacian equals (Kirchhoff's matrix-tree theorem).
Refutation
By [L1], for every vertex index the principal cofactor obtained by deleting row and column has determinant . So the value does not depend on a distinguished choice of index.
This is exactly the negation of the false claim, so the claim is refuted.
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
- Richard P. Stanley, MIT 18.314 handout, Theorem 1.8 (standard reference, not scraped)