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.
A weighted graph with two distinct minimum spanning trees
Statement refuted
Every connected weighted graph has a unique minimum spanning tree.
Facts & Assumptions
Given: The cycle with every edge assigned weight .
Deleting any edge of leaves a two-edge spanning tree (Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices, Spanning trees of a graph).
Total tree weight is the sum of edge weights (Real edge-weighted graphs, total tree weight and minimum spanning trees).
Counterexample
Each of the three two-edge spanning trees has total weight .
Every spanning tree of the three-vertex graph has two edges, so no spanning tree has smaller weight.
Thus all three are MSTs, and uniqueness fails.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
14 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
- ISI Bangalore discrete mathematics notes, Minimal spanning trees (standard reference, not scraped)