Classical Distance-Regular Graphs of Negative Type

✍ Scribed by Chih-wen Weng

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 160 KB
• Volume: 76
• Category: Article
• ISSN: 0095-8956
• DOI: 10.1006/jctb.1998.1892

✦ Synopsis

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).

(ii) 1 is the Hermitian forms graph Her &b (d ).

(iii) :=(b&1)Â2, ;=&(1+b d )Â2, and &b is a power of an odd prime.

1999 Academic Press

1. Introduction

Brouwer, Cohen, and Neumaier found that the intersection numbers of most known families of distance-regular graphs could be described in terms of four parameters (d, b, :, ;) [2, pp. ix, 193]. They invented the term classical to describe those graphs for which this could be done. All classical distance-regular graphs with b=1 are classified by Y. Egawa, A. Neumaier, and P. Terwilliger in a sequence of papers (see [2, p. 195] for a detailed description). In the present paper, we focus on the classical distance-regular graphs with b<&1. The following is our main result.

Main 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).

(ii) 1 is the Hermitian forms graph Her &b (d ).

(iii) :=(b&1)Â2, ;=&(1+b d )Â2, and &b is a power of an odd prime.