Leaves in Representation Diagrams of Bipartite Distance-Regular Graphs

Abstract

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Let θ0 > θ1 ··· > θ D denote the eigenvalues of Γ and let q h ij (0 ≤ h, i, j ≤ D) denote the Krein parameters of Γ. Pick an integer h (1 ≤ h ≤ D − 1). The representation diagram Δ = Δ h is an undirected graph with vertices 0,1,...,D. For 0 ≤ i, j ≤ D, vertices i, j are adjacent in Δ whenever i ≠ j and q h ij ≠ 0. It turns out that in Δ, the vertex 0 is adjacent to h and no other vertices. Similarly, the vertex D is adjacent to D − h and no other vertices. We call 0, D the trivial vertices of Δ. Let l denote a vertex of Δ. It turns out that l is adjacent to at least one vertex of Δ. We say l is a leaf whenever l is adjacent to exactly one vertex of Δ. We show Δ has a nontrivial leaf if and only if Δ is the disjoint union of two paths.