Tails of Bipartite Distance-regular Graphs

Abstract

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.

Distance-Regular Graphs with a Relatively Small Eigenvalue Multiplicity

“…Note that α > 0 and β 0. (Note that in [13], [10], they also allow γ = 0 for a tail and a light tail, respectively. Note that for diameter D 3 this case of γ = 0 only occurs if Γ is an antipodal distance-regular graph of diameter D = 3 and θ = −1 ([10, Theorem 4.1(b)]).…”

A characterization of Leonard pairs using the notion of a tail

“…The following is motivated by [4,Definition 5.1]. Definition 4.5 With reference to Assumption 3.6, let (E, F ) = (E i , E j ) denote an ordered pair of distinct primitive idempotents for A.…”

“…The notion was introduced by M.S. Lang [4] and developed further in [3]. Our main result, which is Theorem 5.1 below, can be viewed as an algebraic version of [3,Theorem 1.1].…”

The Q-polynomial idempotents of a distance-regular graph

“…Following [4], we say that E is three-term recurrent (in short TTR) whenever there exists a complex scalar β such that …”