Graph automorphism

Let $Γ$ be a general graph. A graph automorphism $𝜙 \in A u t (Γ)$ is a bijection $V (𝜙) : V (Γ) \to V (Γ)$ which leaves the adjacency matrix of $Γ$ fully invariant, #m/def/graph i.e.

| Γ (𝑣, 𝑤) | = | Γ (𝜙 (𝑣), 𝜙 (𝑤)) |

for all $𝑣, 𝑤 \in V (Γ)$ . Clearly $A u t (Γ)$ forms a group under composition, which in addition to an action on $V (Γ)$ has an action on $A (Γ)$ . A digraph is called

vertex-transitive iff $A u t (Γ)$ acts transitively on $V (Γ)$ ;
arc-transitive iff $A u t (Γ)$ acts transitively on $A (Γ)$ .

Results

Every arc-transitive digraph is an orbital digraph

#state/tidy | #lang/en | #SemBr