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
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.
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References
E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin-Cummings Lecture Note Ser. 58, Benjamin-Cummings, Menlo Park, CA. 1984.
A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer, New York, 1989.
C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
A. Jurišić, J. Koolen, and P. Terwilliger, “Tight distance-regular graphs,” submitted.
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Pascasio, A.A. Tight Graphs and Their Primitive Idempotents. Journal of Algebraic Combinatorics 10, 47–59 (1999). https://doi.org/10.1023/A:1018624103247
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DOI: https://doi.org/10.1023/A:1018624103247