On local structure of a distance-regular graph of Hamming type
β Scribed by Kazumasa Nomura
- Publisher
- Elsevier Science
- Year
- 1989
- Tongue
- English
- Weight
- 248 KB
- Volume
- 47
- Category
- Article
- ISSN
- 0095-8956
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In this paper, we consider a bipartite distance-regular graph = (X, E) with diameter d β₯ 3. We investigate the local structure of , focusing on those vertices with distance at most 2 from a given vertex x. To do this, we consider a subalgebra R = R(x) of Mat X (C), where X denotes the set of vertice
We prove the following theorem. Theorem. Let 1=(X, R) denote a distance-regular graph with classical parameters (d, b, :, ;) and d 4. Suppose b<&1, and suppose the intersection numbers a 1 {0, c 2 >1. Then precisely one of the following (i) (iii) holds. (i) 1 is the dual polar graph 2 A 2d&1 (&b).
Let denote a distance-regular graph with diameter D 3. Assume has classical parameters (D, b, Ξ±, Ξ²) with b < -1. Let X denote the vertex set of and let A β Mat X (C) denote the adjacency matrix of . Fix x β X and let A \* β Mat X (C) denote the corresponding dual adjacency matrix. Let T denote the s
i=0 the polynomials involved are orthogonal and we display the orthogonality relations. We also show that each of the sequences satisfy a three-term recurrence and a relation known as the Askey-Wilson duality. We then turn our attention to two more bases for W. We find the matrix representations of