Pseudo 1-homogeneous distance-regular graphs

Let be a distance-regular graph of diameter d ≥ 2 and a1 = 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 ciσi−1 + aiσi + biσi+1 = θσi for all i ∈ {0, . . . , d−1}. Furthermore, a pseudo primitive idempotent for θ is Eθ = s ∑di=0 σiAi , where s is any nonzero scalar. Let v̂ be the characteristic vector of a vertex v ∈ V . For an edge xy of 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 Ex̂, Eŷ and Ew is contained in Span{ẑ | z ∈ V , ∂(z, x) = d = ∂(z, y)}. When an edge of is tight with respect to two distinct real numbers, a parameterization with d + 1 parameters of the members of the intersection array of is given (using the pseudo cosines σ1, . . . , σd , and an auxiliary parameter ε). Let S be the set of all the vertices of that are not at distance d from both vertices x and y that are adjacent. The graph 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 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 . Then the graph 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, . . . , σd, ε and the local graph of x is strongly regular with nontrivial eigenvalues a1σ/(1 + σ) and (σ2 − 1)/(σ − σ2). A. Jurišić ( ) Faculty of Computer and Informatic Sciences, University of Ljubljana, Ljubljana, Slovenia e-mail: [email protected] P. Terwilliger Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1388, USA 510 J Algebr Comb (2008) 28: 509–529