Taut distance-regular graphs of even diameter

Abstract

Let G denote a bipartite distance-regular graph with diameter DX4; valency kX3; and Bose-Mesner algebra M: Let y 0 4y 1 4?4y D denote the distinct eigenvalues for G; and for 0pipD; let E i denote the primitive idempotent of M associated with y i : We refer to E 0 and E D as the trivial idempotents of M: Let E and F denote primitive idempotents of M: We say the pair E; F is taut whenever (i) E; F are nontrivial, and (ii) the entry-wise product E3F is a linear combination of two distinct primitive idempotents of M: If G is 2-homogeneous in the sense of Nomura and Curtin, then G has at least one taut pair of primitive idempotents. We define G to be taut whenever G has at least one taut pair of primitive idempotents but G is not 2-homogeneous. Let y denote an eigenvalue of G other than y 0 ; y D ; and let s 0 ; s 1 ; y; s D denote the cosine sequence associated with y: By a result of Curtin, the following are equivalent: (i) G is 2-homogeneous and yAfy 1 ; y DÀ1 g; (ii) there exists a complex scalar l such that s iÀ1 À ls i þ s iþ1 ¼ 0 for 1pipD À 1: Expanding on this, we show that for D even, the following are equivalent: (i) G is taut or 2-homogeneous and yAfy 1 ; y DÀ1 g; (ii) there exists a complex scalar l such that s iÀ1 À ls i þ s iþ1 ¼ 0 for i odd, 1pipD À 1: Using this result, we show that for D even, G is taut or 2-homogeneous if and only if the intersection numbers of G are given by certain rational expressions involving D=2 independent variables. We discuss the known examples of taut distance-regular graphs with even diameter. r

Taut distance-regular graphs and the subconstituent algebra

“…Using (19) and recursion, we routinely find that p 2 (θ) = (θ 2 − b 2 )/c 2 . Using this fact and setting i = 2 in (23), we obtain the desired result. ✷…”

“…Using the equation on the right in ( 16), we routinely verify that θ * 0 = θ * 2 . To obtain (23), set λ = θ in Lemma 4.1(ii), and simplify the result using ( 6), (15), and (16). ✷…”

“…Assume for the moment that Γ is taut. It turns out that the structure of Γ depends to a large extent on the parity of D. We investigated this structure for D even in [23], and for D odd in [24]. In this paper we obtain three characterizations of the taut condition, two of which require the assumption that D is odd.…”

Leaves in Representation Diagrams of Bipartite Distance-Regular Graphs

“…Finally, suppose (h, l) = (D − d, 1). By [7, Theorem 6.2, Corollary 6.3], E D−d • E 1 ∈ Span {E R , E S } for some R, S such that 1 < S < R. But this contradicts (6).…”

“…We refer the reader to the articles of MacLean [5][6][7], Pascasio [9][10][11], and the present author [4] for work on this theme. In this paper we consider the case where E • F is a linear combination of F and one other primitive idempotent.…”