Abstract

Let Γ denote a bipartite distance-regular graph with diameter D≥4, valency k≥3 and intersection numbers c_i, . By a pseudo cosine sequence of Γ we mean a sequence of scalars σ₀,…,σ_D such that σ₀=1 and c_iσ_i−1+b_iσ_i+1=kσ₁σ_i for 1≤i≤D−1. By an associated pseudo primitive idempotent we mean a nonzero scalar multiple of the matrix , where A₀,…,A_D are the distance matrices of Γ. Our main result is the following:

Let σ₀,…,σ_D 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 σ_i−1−βσ_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 c₂≥2.

Article Outline

1. Introduction

2. Distance-regular graphs

3. The almost 2-homogeneous property

4. Pseudo cosine sequences and idempotents

5. Results

Acknowledgements

References