Alphabeta MathGraphs, Walks and Connectivity — Examples
About this page
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.
✓ 8 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 8 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Combinatorics
Real Analysis
2 · Logical flowchart
3 · Items
Example
A small graph separates walks, trails, paths, closed walks and cycles by explicit vertex lists
Example
Connected components, finite within-component distances, undefined cross-component distance and girth are computed in a disconnected graph
Example
K₅ and K_3,3 illustrate complete and complete bipartite graphs, degrees and edge counts
Example
The Petersen graph has ten vertices, is cubic, has girth five and has κ=λ=3
Counterexample
C₆ and the disjoint union of two triangles have the same degree sequence but are not isomorphic
Counterexample
Two triangles sharing one vertex form a connected simple graph with all degrees even that is not a cycle
Counterexample
Two dense lobes meeting at one cut vertex give κ(G)=1<λ(G)=2<δ(G)=3
Example