Some Notes on Distance-Transitive and Distance-Regular Graphs

These are notes from lectures given in the Queen Mary Combinatorics Study 1 Introductory Definitions In these notes, Γ = (V Γ, EΓ) denotes a graph, as we will use G to denote a group. All graphs considered will be simple, finite, connected and undirected. Definition An automorphism of Γ is a bijective function g : V Γ → V Γ such that v ∼ w if and only if g(v) ∼ g(w). The set of all automorphisms is the automorphism group of Γ, denoted by Aut(Γ). If, for all u, v ∈ V Γ, there exists some g ∈ Aut(Γ) such that g(u) = v, the Γ is vertex-transitive. Examples of vertex-transitive graphs include the n-circuits (not very exciting), and (slightly more exciting) the Petersen graph, pictured below. u u u u Figure 1: The 4-circuit (left) and Petersen graph (right)