Leaves in Representation Diagrams of Bipartite Distance-Regular Graphs |
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Authors: | Michael S Lang |
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Institution: | (1) Mathematics Department, Bradley University, 1501 W. Bradley Ave., Peoria, IL, 61625 |
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Abstract: | Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Let 0 > 1 ··· >
D
denote the eigenvalues of and let q
h
ij (0 h, i, j D) denote the Krein parameters of . Pick an integer h (1 h D – 1). The representation diagram =
h
is an undirected graph with vertices 0,1,...,D. For 0 i, j D, vertices i, j are adjacent in whenever i j and q
h
ij 0. It turns out that in , 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 . Let l denote a vertex of . It turns out that l is adjacent to at least one vertex of . We say l is a leaf whenever l is adjacent to exactly one vertex of . We show has a nontrivial leaf if and only if is the disjoint union of two paths. |
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Keywords: | primitive idempotent eigenvalue association scheme Q-polynomial antipodal |
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