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Journal of Algebraic Combinatorics

, Volume 28, Issue 4, pp 509–529 | Cite as

Pseudo 1-homogeneous distance-regular graphs

  • Aleksandar JurišićEmail author
  • Paul Terwilliger
Article

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 σ 0,…,σ d such that σ 0=1 and c i σ i−1+a i σ i +b i σ i+1=θ σ i for all i∈{0,…,d−1}. Furthermore, a pseudo primitive idempotent for θ is E θ =s ∑ i=0 d σ i A i , where s is any nonzero scalar. Let \(\hat{v}\) be the characteristic vector of a vertex vVΓ. 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 θk 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 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 σ 1,…,σ d , 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 σ 1,…,σ d ,ε and the local graph of x is strongly regular with nontrivial eigenvalues a 1 σ/(1+σ) and (σ 2−1)/(σσ 2).

Keywords

Distance-regular graphs Primitive idempotents Cosine sequence Locally strongly regular 1-homogeneous property Tight distance-regular graph Pseudo primitive idempotent Tight edges Pseudo 1-homogeneous 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Faculty of Computer and Informatic SciencesUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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