The local eigenvalues of a bipartite distance-regular graph.

We consider a bipartite distance-regular graph Γ with vertex set X , diameter D ≥ 4 , and valency k ≥ 3 . For 0 ≤ i ≤ D , let Γ i ( x ) denote the set of vertices in X that are distance i from vertex x . We assume there exist scalars r , s , t ∈ R , not all zero, such that r | Γ 1 ( x ) ∩ Γ 1 ( y ) ∩ Γ 2 ( z ) | + s | Γ 2 ( x ) ∩ Γ 2 ( y ) ∩ Γ 1 ( z ) | + t = 0 for all x , y , z ∈ X with path-length distances ∂ ( x , y ) = 2 , ∂ ( x , z ) = 3 , ∂ ( y , z ) = 3 . Fix x ∈ X , and let Γ 2 2 denote the graph with vertex set X ̃ = { y ∈ X ∣ ∂ ( x , y ) = 2 } and edge set R ̃ = { y z ∣ y , z ∈ X ̃ , ∂ ( y , z ) = 2 } . We show that the adjacency matrix of the local graph Γ 2 2 has at most four distinct eigenvalues. We are motivated by the fact that our assumption above holds if Γ is Q -polynomial. [ABSTRACT FROM AUTHOR]