Abstract :
Let Γ be a finite simple undirected graph. An automorphism σ ∈ Aut Γ is an adjacency automorphism of Γ if dist(x,σ(x)) ⩽ 1 for every vertex x ∈ Γ. A graph Γ is adjacency-transitive if for every pair of vertices x, y ∈ V(Γ) there exists a sequence of adjacency automorphisms σ1, σ2, …, σk ∈ Aut Γ such that σ1 σ2 ··· σk(x) = y. Examples of such graphs include certain classes of connected Cayley graphs, but not all of them.
Some basic properties and examples of adjacency-transitive graphs are given and those of valency 3 and 4 are classified.
