Tails of Bipartite Distance-regular Graphs

✍ Scribed by Michael S. Lang

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 205 KB
• Volume: 23
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.2002.0611

✦ Synopsis

Let denote a bipartite distance-regular graph with diameter D ≥ 4 and valency k ≥ 3. Let θ 0 > θ 1 > • • • > θ D denote the eigenvalues of and let E 0 , E 1 , . . . , E D denote the associated primitive idempotents. Fix s (1 ≤ s ≤ D -1) and abbreviate E := E s . We say E is a tail whenever the entrywise product E • E is a linear combination of E 0 , E and at most one other primitive idempotent of . Let q h i j (0 ≤ h, i, j ≤ D) denote the Krein parameters of and let denote the undirected graph with vertices 0, 1, . . . , D where two vertices i, j are adjacent whenever i = j and q s i j = 0. We show E is a tail if and only if one of (i)-(iii) holds: (i) is a path; (ii) has two connected components, each of which is a path; (iii) D = 6 and has two connected components, one of which is a path on four vertices and the other of which is a clique on three vertices.