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In this paper, we prove the following two theorems.Theorem 1 Let Γ denote a distance-regular graphwith diameter d ≥ 3. Suppose E and F are primitive idempotents of Γ, with cosine sequences σ_0, σ_1, …, σ_d and ρ_0, ρ_1, …, ρ_d, respectively. Then the following are equivalent.(i) The entry-wise product E F is a scalar multiple of a primitive idempotent of Γ.(ii) There exists a real number ε such thatLet Γ denote a distance-regular graph with diameter d≥ 3 and eigenvalues &thetas;_0 &thetas;_1 … &thetas;_d. Then Jurišić, Koolen and Terwilliger proved that the valency k andthe intersection numbers a_1, b_1 satisfy They defined Γ to be tight whenever Γ is not bipartite, and equality holds above.Theorem 2 Let Γ denote a distance-regular graph with diameter d ≥ 3 and eigenvalues &thetas;_0 thetas;_1 … &thetas;_d. Let E and F denote nontrivial primitive idempotents of Γ.(i) Suppose Γ is tight. Then E, F satisfy (i), (ii) in Theorem 1 if and only if E, F are a permutation of E_1, E_d.(ii) Suppose Γ is bipartite. Then E, F satisfy (i), (ii) in Theorem 1 if and only if at least one of E, F is equal to E_d.(iii) Suppose Γ is neither bipartite nor tight. Then E, F never satisfy (i), (ii) in Theorem 1.

Tight Graphs and Their Primitive Idempotents