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Pseudo 1-homogeneous distance-regular graphs. (English) Zbl 1160.05060
Summary: Let $$\Gamma$$ be a distance-regular graph of diameter $$d\geq 2$$ and $$a_{1} \neq 0$$. Let $$\theta$$ be a real number. A pseudo cosine sequence for $$\theta$$ is a sequence of real numbers $$\sigma_{0},\dots ,\sigma_{d}$$ 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 $$i\in \{0,\dots ,d - 1\}$$. Furthermore, a pseudo primitive idempotent for $$\theta$$ is $$E_{\theta }=s \sum_{i=0}^ d \sigma_{i} A_{i}$$, where $$s$$ is any nonzero scalar. Let $$\hat{v}$$ 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 $$\theta \neq k$$ and a nontrivial linear combination of vectors $$E\hat{x} , E\hat{y}$$ and $$Ew$$ is contained in $$\mathrm{Span}\{\hat{z}\mid z\in V{\Gamma},\;\partial(z,x)=d=\partial(z,y)\}$$. 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 $$\sigma_{1},\dots ,\sigma_{d}$$, 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 $$\sigma _{1},\dots ,\sigma_{d },\varepsilon$$ and the local graph of $$x$$ is strongly regular with nontrivial eigenvalues $$a_{1} \sigma /(1+\sigma )$$ and $$(\sigma_{2} - 1)/(\sigma - \sigma_{2})$$.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C12 Distance in graphs
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##### References:
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