Pseudo 1-homogeneous distance-regular graphs

Let Γ 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 forθ is Eθ=s ∑i=0dσiAi, where s is any nonzero scalar. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{v}$\end{document} be the characteristic vector of a vertex v∈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 toθ whenever θ≠k and a nontrivial linear combination of vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E\hat{x}$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E\hat{y}$\end{document} and Ew is contained in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{Span}\{\hat{z}\mid z\in V{\Gamma},\ \partial(z,x)=d=\partial(z,y)\}$\end{document} . 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).