Tight Distance-regular Graphs and the Subconstituent Algebra

Abstract

We consider a distance-regular graph with diameter D ≥ 3, intersection numbers a i , b i , c i and eigenvalues k = θ 0 > θ 1 > · · · > θ D . Let X denote the vertex set of and fix x ∈ X. Let T = T (x) denote the subalgebra of Mat X (C) generated by A, E , where E i denotes the primitive idempotent of A associated with θ i . We show this basis is orthogonal (with respect to the Hermitean dot product) and we compute the square norm of each basis vector. We show, where A i denotes the ith distance matrix for . We find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square norm of each basis vector. We find the transition matrix relating our two bases for W . For notational convenience, we say is 1-thin with respect to x whenever every irreducible T -module with endpoint 1 is thin. Similarly, we say is tight with respect to x whenever every irreducible T -module with endpoint 1 is thin with local eigenvalueθ 1 orθ D . In [J. Algebr. Comb., 12, (2000), Jurišić, Koolen and Terwilliger showedThey defined to be tight whenever is nonbipartite and equality holds above. We show the following are equivalent: (i) is tight; (ii) is tight with respect to each vertex; (iii) is tight with respect to at least one vertex. We show the following are equivalent: (i) is tight; (ii) is nonbipartite, a D = 0, and is 1-thin with respect to each vertex; (iii) is nonbipartite, a D = 0, and is 1-thin with respect to at least one vertex.

Pseudo 1-homogeneous distance-regular graphs

“…Tight graphs have been characterized in a number of interesting ways. See Jurišić, Koolen and Terwilliger [7], Pascasio [8] and Go & Terwilliger [4]. We collect some of their results in the following theorem.…”

“…called the Fundamental Bound, and was defined to be tight whenever it is not bipartite, and equality holds in (4). Tight graphs have been characterized in a number of interesting ways.…”

The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph

“…Thus it is advantageous to choose a base subset rather than a base vertex. In fact the results in [23,48,49] are those on T ( 1 (x))-modules. Since w( 1 (x)) = 2, our results are generalization to the case with arbitrary width.…”

“…In Section 11, we give description of the case when the width w(Y ) of Y is at most 2. This contains the case when Y = 1 (x), which was studied in [23,48,49].…”

“…The proofs of (ii), (iii), and (iv) are very similar to those of Lemma 3.2 and therefore omitted. See [23,48] …”