DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

Graph automorphisms and group actions on a simplicial graph

Definition

Let $X = (V, E)$ be an oriented graph. A graph automorphism of $X$ is a pair of bijections on $V$ and $E$ , both denoted by $g$ , such that

$g (o (e)) = o (g (e)), g (t (e)) = t (g (e)), g (\overset{ˉ}{e}) = \overline{g (e)} .$

A group action on the simplicial graph $X$ is a left action (Left group actions, transitive actions, and faithful actions) of a group $G$ on both $V$ and $E$ such that each $g \in G$ acts by a graph automorphism in the above sense.

Depends on

Used by

Edge inversions and actions without inversions Definition
Fixed subtrees and minimal invariant subtrees Definition

Dependency tree · two levels

3 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

Jean-Pierre Serre, Trees (standard reference, not scraped)
Yuriy Tumarkin, Groups Acting on Trees (standard reference, not scraped)