Bipartite Distance-regular Graphs, Part II

This is a continuation of ``Bipartite Distance-regular Graphs, Part I''. We continue our study of the Terwilliger algebra T of a bipartite distance-regular graph. In this part we focus on the thin irreducible T-modules of endpoint 2 and on those distance-regular graphs for which every irreducible T-module of endpoint 2 is thin. 8. Introduction to Part II Let G ˆ …X ;E† denote a bipartite distance-regular graph with diameter DV 4 (formal de®nitions appear in Part I of this paper), and ®x a vertex x of G . The Terwilliger algebra T ˆ T…x† of G with respect to x is the subalgebra of MatX …C† generated by A, E 0 ;E 1 ; . . . ;E D, where A is the adjacency matrix for G, and where E i denotes the projection onto the i th subconstituent of G with respect to x. An irreducible T-module W is said to be thin whenever dim E i W U 1 for 0U iUD. The endpoint of W is minfi jE i W 0 0g. The diameter of W is jfi j 0U iUD;E i W 0 0gj ÿ 1. In Part I we studied the irreducible T-modules of endpoint 0 and 1 and showed that their structure is fairly simple. In Part II we examine the irreducible T-modules of endpoint 2. The situation here is somewhat more complicated. Let v denote any nonzero vector in the second subconstituent of the standard module which is orthogonal to all irreducible T-modules of endpoint 0 and 1, and put vi ˆ E i Aiÿ2v, vi ˆ E i Ai‡2v. We ®rst describe the action of T on the vi and vi (Theorem 9.4). Let W denote an irreducible T-module of endpoint 2. We show that the diameter of W is one of Dÿ 2, Dÿ 3, Dÿ 4 (Theorem 10.1). Assume W is thin. We show W has orthogonal basis v2 , v ‡ 3 , . . ., v ‡ d‡2, where d denotes the diameter of W (Lemma 10.2(i)). We describe the action of T on this basis of W in terms of the intersection numbers of G and one additional real parameter c ˆ c…W†, which we * This paper is based upon research supported under a National Science Foundation Graduate Research Fellowship. Sections 1-7 appeared in Part I, Graphs and Combinatorics 15: 143±158 refer to as the type of W (Lemma 10.8). We show the isomorphism class of W is determined by c (Lemma 10.10). We now assume each irreducible T-module of endpoint 2 is thin and obtain the following two-fold result. First, we present a recurrence giving the intersection numbers of G in terms of the diameter D of G and the set of ordered pairs f…c;mult…c†† jc A F2g, where F2 denotes the set of distinct types of irreducible Tmodules with endpoint 2, and where mult…c† denotes the multiplicity with which the module of type c appears in the standard module (Theorem 13.1). Secondly, we show that the set of ordered pairs f…c;mult…c†† jc A F2g is determined by the intersection numbers k, b2, b3 of G and the spectrum of the graph G 2 2 ˆ … ~ X ; ~ E†, where ~ X ˆ fy A X j q…x; y† ˆ 2g, ~ E ˆ fyz j y; z A ~ X ; q…y; z† ˆ 2g (Theorem 11.7). We conclude that if every irreducible T-module of endpoint 2 is thin, then the intersection numbers of G are determined by the diameter D of G, the intersection numbers k, b2, b3 of G , and the spectrum of G 2 2 (Corollary 13.2). This project is part of our study of the Q-polynomial property. It is known that if G is Q-polynomial and bipartite, then every irreducible T-module is thin and jF2jU 2 [5, Corollary 4.11], [6, Corollary 5.7]. 9. The Vectors vi , v ÿ i Let G denote a bipartite distance-regular graph as in De®nition 2.2. We now begin our investigation of the space j2V . Recall that in Section 7 we studied vectors E i Aiÿ1v and E i Ai‡1v for v A E 1 j1V in order to understand the irreducible Tmodules of endpoint 1. Now let W denote any irreducible T-module of endpoint 2, and pick any nonzero v A E 2 W . We consider the vectors v ‡ i ˆ E i Aiÿ2v and vi ˆ E i Ai‡2v. Lemma 9.1. With the notation of De®nition 2.2, assume G is bipartite. Then for all integers i …0U iUD† (i) AiE 2 ˆ E iÿ2AiE 2 ‡ E i AiE 2 ‡ E i‡2AiE 2 ; …57† (ii) E i JE 2 ˆ E i Aiÿ2E 2 ‡ E i AiE 2 ‡ E i Ai‡2E 2 ; …58† (iii) AiE 2 ˆ E i‡2AiE 2 ÿ E i Ai‡2E 2 ‡ AiE 0 JE 2 : …59† Proof. Similar to that of Lemma 7.1. r De®nition 9.2. With the notation of De®nition 2.2, assume G is bipartite, let j2 be as in De®nition 3.7, and pick any nonzero v A j2E 2 V . For all integers i, de®ne vi ˆ E i Aiÿ2v; vi ˆ E i Ai‡2v: …60† Observe by (17) that v2 ˆ v, vi ˆ 0 if i < 2 or i > D, and vi ˆ 0 if i < 2 or i > Dÿ 2. Corollary 9.3. With the notation of De®nition 2.2, assume G is bipartite. 378 B. Curtin