Let ▫$\Gamma$▫ be a distance-regular graph of diameter ▫$d \ge 2$▫ and ▫$a_1 \ne 0$▫. Let ▫$\theta$▫ be a real number. A pseudo cosine sequence for ▫$\theta$▫ is a sequence of real numbers ...▫$\sigma_0,...,sigma_d$▫ such that ▫$\sigma_0=1$▫ and ▫$c_i \sigma_{i-1}+a_i \sigma_i + b_i\sigma_{i+1} = \theta \sigma_i$▫ for all ▫$i \in \{0,...,d-1\}$▫. Furthermore, a pseudo primitive idempotent for ▫$\theta$▫ is ▫$E_\theta = s\sum_{i=0}^d \sigma_iA_i$▫, where ▫$s$▫ is any nonzero scalar. Let ▫$\widehat{v}$▫ be the characteristic vector of a vertex ▫$v in V\Gamma$▫. For an edge ▫$xy$▫ of ▫$\Gamma$▫ 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 ▫$\theta$▫ whenever ▫$\theta \ne k$▫ and a nontrivial linear combination of vectors ▫$E\widehat{x}$▫, ▫$E\widehat{y}$▫ and ▫$E\widehat{w}$▫ is contained in Span▫$\{\widehat{z}| z \in V\Gamma, \; \partial(z,x) = \partial(x,y)\}$▫. When an edge of ▫$\Gamma$▫ is tight with respect to two distinct real numbers, a parameterization with ▫$d+1$▫ parameters of the members of the intersection array of ▫$\Gamma$▫ is given (using the pseudo cosines ▫$\sigma_1,...,\sigma_d$▫, and an auxiliary parameter ▫$\varepsilon$▫). Let ▫$S$▫ be the set of all the vertices of ▫$\Gamma$▫ that are not at distance ▫$d$▫ from both vertices ▫$x$▫ and ▫$y$▫ that are adjacent. The graph ▫$\Gamma$▫ 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 ▫$\Gamma$▫ 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 ▫$\Gamma$▫. Then the graph ▫$\Gamma$▫ 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 ▫$\sigma_1, ... ,\sigma_d, \varepsilon$▫ and the local graph of ▫$x$▫ is strongly regular with nontrivial eigenvalues ▫$a_1\sigma/(1+\sigma)$▫ and ▫$(\sigma_2 - 1)/(\sigma - \sigma_2)$▫.
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