Pseudo 1-homogeneous distance-regular graphs |
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Authors: | Aleksandar Jurišić Paul Terwilliger |
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Institution: | (1) Faculty of Computer and Informatic Sciences, University of Ljubljana, Ljubljana, Slovenia;(2) Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1388, USA |
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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
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 to
θ whenever θ≠k and a nontrivial linear combination of vectors
,
and Ew is contained in
. 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)/(σ−σ
2). |
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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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