DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

Locally finite graphs and vertex degree without a finiteness hypothesis

Definition

Let $G$ be a simple graph and let $v \in V (G)$ . Its neighbourhood is

$N_{G} (v) : = {w \in V (G) : {v, w} \in E (G)} .$

If $N_{G} (v)$ is finite, its cardinality is the degree of $v$ and is written $\deg_{G} (v) : = ∣ N_{G} (v) ∣ .$

The graph $G$ is locally finite if $N_{G} (v)$ is finite for every vertex $v$ . Equivalently, every vertex has finite degree.

Depends on

Used by

Dependency tree · two levels

11 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)