DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-01

How 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.

The spanning-tree number $τ (G)$

Definition

For a finite graph $G$ , its spanning-tree number is

$τ (G) : = ∣ {T : T is a spanning tree of G} ∣ .$

This is a natural number because the displayed set is finite (The set of spanning trees of a finite graph is finite, The cardinality $∣ A ∣$ of a finite set). In particular, $τ (G) = 0$ exactly when $G$ has no spanning tree.

Depends on

Used by

τ(Cₙ)=n for every n≥3 Example
τ(K₄)=16 by direct Prüfer enumeration and Cayley's formula Example
Cayley's formula: τ(Kₙ)=nⁿ⁻² for n≥2, with τ(K₁)=1 and τ(K₀)=0 Theorem
Kirchhoff's matrix-tree theorem Theorem

Dependency tree · two levels

13 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

Reinhard Diestel, Graph Theory, Preview Chapter 1 (standard reference, not scraped)
ISI Bangalore discrete mathematics notes, Trees and Cayley’s theorem (standard reference, not scraped)

The spanning-tree number τ(G)τ(G)