Pseudo primitive idempotents and almost 2-homogeneous bipartite distance-regular graphs

Abstract

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 4 , valency k ≥ 3 and intersection numbers c i , b i ( 0 ≤ i ≤ D ) . By a pseudo cosine sequence of Γ we mean a sequence of scalars σ 0 , … , σ D such that σ 0 = 1 and c i σ i − 1 + b i σ i + 1 = k σ 1 σ i for 1 ≤ i ≤ D − 1 . By an associated pseudo primitive idempotent we mean a nonzero scalar multiple of the matrix ∑ i = 0 D σ i A i , where A 0 , … , A D are the distance matrices of Γ . Our main result is the following: Let σ 0 , … , σ D denote a pseudo cosine sequence of Γ with σ 1 ∉ { − 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 σ i − 1 − β σ i + σ i + 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 c 2 ≥ 2 .