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.
Spanning trees of a graph
Definition
Let be a graph. A spanning tree of is a spanning subgraph of that is a tree (Subgraphs, induced subgraphs and spanning subgraphs, Trees, forests, leaves and isolated vertices). Equivalently,
and is connected and acyclic.
The null graph has no spanning tree under the library convention that a tree is nonempty and connected.
Depends on
Used by
- A weighted graph with two distinct minimum spanning trees Counterexample
- Real edge-weighted graphs, total tree weight and minimum spanning trees Definition
- A connected graph with two distinct spanning trees Example
- A fundamental cycle and a fundamental cut in a fixed spanning tree Example
- τ(Cₙ)=n for every n≥3 Example
- Every edge of a connected graph lies in every spanning tree False statement
- A maximal acyclic spanning subgraph of a connected graph is a spanning tree Lemma
- Every edge of a spanning tree determines a fundamental cut, and every edge crossing it restores a spanning tree Lemma
- Every edge outside a spanning tree determines a unique fundamental cycle Lemma
- The set of spanning trees of a finite graph is finite Lemma
- The spanning-tree exchange lemma Lemma
- A finite graph is connected if and only if it has a spanning tree Theorem
- Prim's growing-tree procedure produces a minimum spanning tree Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 9 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Reinhard Diestel, Graph Theory, Preview Chapter 1 (standard reference, not scraped)