Let Γ denote a bipartite distance-regular graph with diameter D≥4, valency k≥3 and intersection numbers ci, . 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 , 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{-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.
展开▼
