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.
Cycles, trees and forests in a simple graph on an arbitrary vertex set
Definition
Let be a simple graph.
A cycle in is a cycle in the sense of Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges, that is, a closed walk of length at least whose vertices are distinct except for the repeated endpoint.
The graph is acyclic if it contains no cycle.
A tree is a connected acyclic simple graph.
A forest is an acyclic simple graph.
Depends on
Used by
- The Cayley graphs of ℤ/2 for {1} and of ℤ for {-1,1} are trees, and neither generating set is free Counterexample
- The Cayley graph of the free group on two generators is the tree in which every vertex has four neighbours Example
- A nonempty simple graph is a tree if and only if each pair of vertices is joined by exactly one path Theorem
- If no product of two members of a generating set is the identity and the Cayley graph is a tree, the set is a free basis Theorem
- The Cayley graph of a free group with respect to a free basis is a tree Theorem
Dependency tree · two levels
6 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
- C. Loh, Geometric Group Theory: An Introduction (2015 course version), Section 3.1 (standard reference, not scraped)