An Inequality Involving the Local Eigenvalues of a Distance-Regular Graph

Let denote a distance-regular graph with diameter D ≥ 3, valency k, and intersection numbers ai , bi , ci . Let X denote the vertex set of and fix x ∈ X . Let denote the vertex-subgraph of induced on the set of vertices in X adjacent x . Observe has k vertices and is regular with valency a1. Let η1 ≥ η2 ≥ · · · ≥ ηk denote the eigenvalues of and observe η1 = a1. Let denote the set of distinct scalars among η2, η3, . . . , ηk . For η ∈ let multη denote the number of times η appears among η2, η3, . . . , ηk . Let λ denote an indeterminate, and let p0, p1, . . . , pD denote the polynomials in R[λ] satisfying p0 = 1 and λpi = ci+1 pi+1 + (ai − ci+1 + ci )pi + bi pi−1 (0 ≤ i ≤ D − 1), where p−1 = 0. We show 1 + ∑ η∈ η =−1 pi−1(η̃) pi (η̃)(1 + η̃) multη ≤ k bi (1 ≤ i ≤ D − 1), where we abbreviate η̃ = −1−b1(1+η)−1. Concerning the case of equality we obtain the following result. Let T = T (x) denote the subalgebra of MatX (C) generated by A, E∗ 0 , E ∗ 1 , . . . , E ∗ D , where A denotes the adjacency matrix of and E∗ i denotes the projection onto the i th subconstituent of with respect to x . T is called the subconstituent algebra or the Terwilliger algebra. An irreducible T -module W is said to be thin whenever dimE∗ i W ≤ 1 for 0 ≤ i ≤ D. By the endpoint of W we mean min{i |E∗ i W = 0}. We show the following are equivalent: (i) Equality holds in the above inequality for 1 ≤ i ≤ D − 1; (ii) Equality holds in the above inequality for i = D − 1; (iii) Every irreducible T -module with endpoint 1 is thin.