Tight Distance-regular Graphs and the Subconstituent Algebra

We consider a distance-regular graph Γ with diameter D ≥ 3, intersection numbers ai, bi, ciand eigenvalues k = θ0 > θ1 > ⋯ > θD. Let X denote the vertex set of Γ and fix x ∈ X. Let T = T(x) denote the subalgebra of Mat X(C) generated by A, E0 * , E1 * ,⋯ , ED * , where A denotes the adjacency matrix of Γ and Ei * denotes the projection onto the i th subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T -module W is said to be thin whenever dimEi * W ≤ 1 for 0 ≤ i ≤ D. By the endpoint of W we mean min{ i | Ei * W ≠ = 0}. Let W denote a thin irreducible T -module with endpoint 1. Observe E1 * W is a one-dimensional eigenspace for E1 * AE1 * ; let η denote the corresponding eigenvalue. We call η the local eigenvalue of W. It is known yeujc5970x.gif θ1 ≤ η ≤ yeujc5971x.gif θDwhere yeujc5972x.gif θ1 = − 1 − b1(1 + θ1) − 1and yeujc5973x.gif θD = − 1 − b1(1 + θD) − 1. Let n = 1 or n = D and assume η = yeujc5974x.gif θn. We show the dimension of W is D − 1. Let v denote a nonzero vector in E1 * W. We show W has a basis Eiv(1 ≤ i ≤ D, i ≠ = n), where Eidenotes the primitive idempotent of A associated with θi. 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 Ei + 1 * Aiv(0 ≤ i ≤ D − 2), where Aidenotes the i th distance matrix for Γ. 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 Γ is 1- thin with respect tox whenever every irreducible T -module with endpoint 1 is thin. Similarly, we say Γ is tight with respect tox whenever every irreducible T -module with endpoint 1 is thin with local eigenvalue yeujc5975x.gif θ1or yeujc5976x.gif θD. In [ J. Algebr. Comb., 12,(2000), 163–197] Jurišić, Koolen and Terwilliger showed (θ1 + ka1 + 1) (θD + ka1 + 1) ≥ − ka1b1(a1 + 1)2.They defined Γ to be tight whenever Γ is nonbipartite and equality holds above. We show the following are equivalent: (i) Γ is tight; (ii) Γ is tight with respect to each vertex; (iii) Γ is tight with respect to at least one vertex. We show the following are equivalent: (i) Γ is tight; (ii) Γ is nonbipartite, aD = 0, and Γ is 1-thin with respect to each vertex; (iii) Γ is nonbipartite, aD = 0, and Γ is 1-thin with respect to at least one vertex.