### Tight Distance-regular Graphs and the Subconstituent Algebra

Junie T. Go, Paul Terwilliger
2002 European journal of combinatorics (Print)
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 f x x ∈ X. where A denotes the adjacency matrix of and E * i denotes the projection onto the ith subconstituent of with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of with respect to x. An irreducible T -module W is said to be thin whenever dim where E i denotes the primitive idempotent
more » ... 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 W has a basis E , where A i denotes the ith distance matrix for . We f nd 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 f nd 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. They def ned 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.