On adjacency-transitive graphs

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 σ 1, σ 2, …, σ k ∈ Aut Γ such that σ 1 σ 2 ··· σ k ( 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.