Pseudo primitive idempotents and almost 2-homogeneous bipartite distance-regular graphs
نویسنده
چکیده
Let Γ denote a bipartite distance-regular graph with diameter D ≥ 4, valency k ≥ 3 and intersection numbers ci , bi (0 ≤ i ≤ D). By a pseudo cosine sequence of Γ we mean a sequence of scalars σ0, . . . , σD such that σ0 = 1 and ciσi−1 + biσi+1 = kσ1σi for 1 ≤ i ≤ D − 1. By an associated pseudo primitive idempotent we mean a nonzero scalar multiple of the matrix ∑D i=0 σi Ai , where A0, . . . , AD are the distance matrices of Γ . Our main result is the following: Let σ0, . . . , σD denote a pseudo cosine sequence of Γ with σ1 6∈ {−1, 1} and let E denote an associated pseudo primitive idempotent. The following are equivalent: (i) the entrywise product of E with itself is a linear combination of the all-ones matrix and a pseudo primitive idempotent of Γ ; (ii) there exists a scalar β such that σi−1−βσi +σi+1 = 0 for 1 ≤ i ≤ D−1. Moreover, Γ has such a pseudo cosine sequence and pseudo primitive idempotent if and only if Γ is almost 2-homogeneous with c2 ≥ 2. c © 2007 Elsevier Ltd. All rights reserved.
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عنوان ژورنال:
- Eur. J. Comb.
دوره 29 شماره
صفحات -
تاریخ انتشار 2008