Tight Graphs and Their Primitive Idempotents

Abstract

In this paper, we prove the following two theorems.Theorem 1 Let Γ denote a distance-regular graph with diameter d ≥ 3. Suppose E and F are primitive idempotents of Γ, 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 Γ.(ii) There exists a real number ∈ such that \(\sigma _i \rho _i - \sigma _{i - 1} \rho _{i - 1} = \in (\sigma _{i - 1} \rho _i - \sigma _i \rho _{i - 1} ) (1 \leqslant i \leqslant d).\) Let Γ denote a distance-regular graph with diameter d ≥ 3 and eigenvalues θ0 > θ1 > ... > θd. Then Jurišić, Koolen and Terwilliger proved that the valency k and the intersection numbers a1, b1 satisfy $$\begin{gathered} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{d} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{s} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{z} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{H} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{h} \bar 1 \hfill \\ \left( {\theta _1 + \frac{k}{{a_1 + 1}}} \right)\left( {\theta _\alpha + \frac{k}{{a_1 + 1}}} \right) \geqslant \frac{{ - ka_1 b_1 }}{{\left( {a_1 + 1} \right)^2 }}. \hfill \\ \end{gathered} $$ 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 θ0 > θ1 > ... > θd. 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.