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.
Unless stated otherwise, graph means finite, simple and undirected; orders, sizes and empty-set conventions are fixed here
For a graph (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets), its order is and its size is (The cardinality of a finite set). The null graph has and . An edgeless graph has but may have vertices. Thus the null graph is the unique graph with no vertices, while an edgeless graph need not be null.
All sums indexed by or use their ordinary empty values. A minimum or maximum taken over the vertex set is used only when ; in particular, minimum and maximum degree are not assigned values for the null graph. Connectivity conventions for the null graph and the one-vertex graph are stated with the definition of connectivity.
Depends on
Used by
- Connected graphs and connected components defined by the existence of vertex paths Definition
- Empty and complete graphs, complete bipartite graphs, and the convention that Pₙ and Cₙ have n vertices Definition
- Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph Definition
- Ordinary-subgraph extremal number ex(n,H), Turán graph T_n,r, and balanced blowup H[s] Definition
- Simple graphs on an arbitrary vertex set Definition
Dependency tree · two levels
9 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
- R. Diestel, Graph Theory, Chapter 1 preview (standard reference, not scraped)