Tight Graphs and Their Primitive Idempotents

In this paper, we prove the following two theorems. Theorem 1 Let 0 denote a distance-regular graph with diameter d ≥ 3. Suppose E and F are primitive idempotents of 0, with cosine sequences σ0, σ1, . . . , σd and ρ0, ρ1, . . . , ρd , respectively. Then the following are equivalent. (i) The entry-wise product E ◦ F is a scalar multiple of a primitive idempotent of 0. (ii) There exists a real number 2 such that σiρi − σi−1ρi−1 = 2(σi−1ρi − σiρi−1) (1 ≤ i ≤ d). Let 0 denote a distance-regular graph with diameter d ≥ 3 and eigenvalues θ0 > θ1 > · · · > θd . Then Juris̆ić, Koolen and Terwilliger proved that the valency k and the intersection numbers a1, b1 satisfy ( θ1 + k a1 + 1 )( θd + k a1 + 1 ) ≥ −ka1b1 (a1 + 1)2 . They defined 0 to be tight whenever 0 is not bipartite, and equality holds above. Theorem 2 Let 0 denote a distance-regular graph with diameter d ≥ 3 and eigenvalues θ0 > θ1 > · · · > θd . Let E and F denote nontrivial primitive idempotents of 0. (i) Suppose 0 is tight. Then E, F satisfy (i), (ii) in Theorem 1 if and only if E, F are a permutation of E1, Ed . (ii) Suppose 0 is bipartite. Then E, F satisfy (i), (ii) in Theorem 1 if and only if at least one of E, F is equal to Ed . (iii) Suppose 0 is neither bipartite nor tight. Then E, F never satisfy (i), (ii) in Theorem 1.