Pseudo 1-homogeneous distance-regular graphs

Abstract

Let Γ be a distance-regular graph of diameter d ≥ 2 and a 1 ≠ 0. Let θ be a real number. A pseudo cosine sequence for θ is a sequence of real numbers σ0,..., σd such that σ0=1 and ci σi-1+ai σi +bi σi+1=θ σi for all i {0,...,d-1}. Furthermore, a pseudo primitive idempotent for θ is Eθ = s ∑i=0d σiAi, where s is any nonzero scalar. Let v̂ be the characteristic vector of a vertex v ε VΓ. For an edge xy of Γ and the characteristic vector w of the set of common neighbours of x and y, we say that the edge xy is tight with respect to θ whenever θ ≠ k and a nontrivial linear combination of vectors E x̂, E ŷ and Ew is contained in Span{ẑ| z ε VΓ, ∂(z, x) = d = ∂(z,y)}. When an edge of Γ is tight with respect to two distinct real numbers, a parameterization with d+1 parameters of the members of the intersection array of Γ is given (using the pseudo cosines σ1,...,σd, and an auxiliary parameter ε). Let S be the set of all the vertices of Γ that are not at distance d from both vertices x and y that are adjacent. The graph Γ is pseudo 1-homogeneous with respect to xy whenever the distance partition of S corresponding to the distances from x and y is equitable in the subgraph induced on S. We show Γ is pseudo 1-homogeneous with respect to the edge xy if and only if the edge xy is tight with respect to two distinct real numbers. Finally, let us fix a vertex x of Γ. Then the graph Γ is pseudo 1-homogeneous with respect to any edge xy, and the local graph of x is connected if and only if there is the above parameterization with d+1 parameters σ1,...,σ d, ε and the local graph of x is strongly regular with nontrivial eigenvalues a1σ/(1+σ) and (σ2-1)/(σ-σ2). © 2008 Springer Science+Business Media, LLC.