DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Label-preserving graph isomorphisms

Definition

Let $G = (V, E, ℓ)$ and $H = (W, F, m)$ be finite vertex-labeled simple undirected graphs with labels in the same set $Σ$ . A label-preserving graph isomorphism from $G$ to $H$ is a bijection $φ : V \to W$ such that $ℓ (v) = m (φ (v))$ for every $v \in V$ , and for all distinct $u, v \in V$ , ${u, v} \in E$ if and only if ${φ (u), φ (v)} \in F$ . The graphs are isomorphic when such a bijection exists, and nonisomorphic otherwise.

Definition references

finite vertex-labeled simple undirected graphs

Morris et al., “Weisfeiler and Leman Go Neural,” §2

Li and Leskovec, Graph Representation Learning, Chapter 5 §5.1

Xu, Hu, Leskovec, and Jegelka, “How Powerful are Graph Neural Networks?”, §2

Depends on

Finite simple undirected graphs with vertex labels

Used by

Dependency tree · one level

1 result within one dependency step 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

Christopher Morris et al., Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks (standard reference, not scraped)
Pan Li and Jure Leskovec, Graph Representation Learning (standard reference, not scraped)
Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka, How Powerful are Graph Neural Networks? (standard reference, not scraped)