The number of distinct spanning trees of a connected graph equals any cofactor of the graph's Laplacian matrix. Proved by Gustav Kirchhoff in the context of electrical circuit theory, it gives an exact algebraic formula for a fundamental graph-counting problem.
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StatementKirchhoff's theorem, or Kirchhoff's matrix tree theorem, is a theorem about the number of spanning trees in a graph. It states that this number can be computed as any cofactor of the graph's Laplacian matrix. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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1. Kirchhoff's Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, opening two sentences
Kirchhoff's theorem or Kirchhoff's matrix tree theorem is a theorem about the number of spanning trees in a graph. It states that this number can be computed as any cofactor of the graph's Laplacian matrix.
lead paragraph, sentence naming the publication year
The theorem is named after the German mathematician Gustav Kirchhoff, who published it in 1847.
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