Let Γ denote a distance-regular graph with diameter D ≥ 3, valency k, and intersection numbers a i, b i, c i. Let X denote the vertex set of Γ and fix x ∈ X. Let Δ denote the vertex-subgraph of Γ induced on the set of vertices in X adjacent X. Observe Δ has k vertices and is regular with valency a 1. Let η1 ≥ η2 ≥ ··· ≥ η k denote the eigenvalues of Δ and observe η1 = a 1. Let Φ denote the set of distinct scalars among η2, η3, ..., η k . For η ∈ Φ let multη denote the number of times η appears among η2, η3,...,η k . Let λ denote an indeterminate, and let p 0, p1, ...,p D denote the polynomials in \(\mathbb{R}\)[λ] satisfying p 0 = 1 andλp i = c i+1 p i+1 + (a i − c i+1 + c i)p i + b i p i−1 (0 ≤ i ≤ D − 1),where p −1 = 0. We show EquationSource % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0de9qqFf0x% c9q8qqaqFn0dXdir-xcvk9ar-Jbba9q8Gq0-yq-He9q8qqQ8frFve9% Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIXaGaey% 4kaSYaaabuaeaadaWcaaqaaiaadcfadaWgaaqcbaAaaiaadMgacqGH% sislcaaIXaaaleqaaOGaaiikaiqbeE7aOzaaiaGaaiykaaqaaiaadc% fadaWgaaqcbaAaaiaadMgaaSqabaGccaGGOaGafq4TdGMbaGaacaGG% PaGaaiikaiaaigdacqGHRaWkcuaH3oaAgaacaiaacMcaaaaalqaabe% qaaiabeE7aOjabgIGiolabfA6agbqaaiabeE7aOjabgcMi5kabgkHi% Tiaaigdaaaqab0GaeyyeIuoakiaabccacaqGTbGaaeyDaiaabYgaca% qG0bWaaSbaaSqaaiabeE7aObqabaGccqGHKjYOdaWcaaqaaiaadUga% aeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaaaakiaabccacaqGOaGaae% ymaiabgsMiJkaadMgacqGHKjYOcaWGebGaeyOeI0IaaGymaiaabMca% caqGSaaaaa!6BFB! where we abbreviate \(\tilde \eta \) = −1 − b 1(1+η)−1. Concerning the case of equality we obtain the following result. Let T = T(x) denote the subalgebra of Mat X (\(\mathbb{C}\)) generated by A, E*0, E*1, ..., E* D , where A denotes the adjacency matrix of Γ and E* i denotes the projection onto the ith subconstituent of Γ 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.