Geometries originating from certain distance-regular graphs

Summary

Distance-regular graphs having intersection number c² = 1 are point graphs of linear incidence systems. This simple observation plays a crucial role in both the existence proof of a regular near octagon “associated with the Hall-Janko group” whose point graph is the unique automorphic graph with intersection array {10, 8, 8, 2; 1, 1, 4, 5} and the non-existence proof of a distance-regular graph with intersection array {12, 8, 6,…; 1, 1, 2,…}. These results imply a partial answer to a problem put forward by Biggs in 1976.

BASIC NOTIONS

Let Γ = (V,E) be a connected graph of diameter d and let Γ_i(α) for α є V denote the set of vertices at distance i from α. We recall from [2] that Γ is distance-regular if for any i (0≤i≤d) the numbers b_i = |Γ_i+1(α) ∩ Γ₁ (β)| and c_i = |Γ_i-1(α) ∩ Γ₁ (β)| do not depend on the choice of α, β such that β є Γ_i(α). Of course, b_d = 0 and c_l = 1. Write k = | Γ₁(α)|. The array {k, b₁, b₂,…, b_d-1; c₁, c₂,…, c_d} is called the intersection array of Γ.

The graph Γ is called distance-transitive if its automorphism group Aut(Γ) is transitive on each of the classes {{α, β} ⊃ V | β ∈Γ_i(α)} (0≤i≤d), and Γ is called automorphic (cf. [3]) whenever it is distance-transitive, not a complete graph or a line graph, and has an automorphism group which is primitive on V.