Some Results on the Eigenvalues of Distance-Regular Graphs

Light tails and the Hermitian dual polar graphs

“…both hold, by(11) This means that equality holds if and only if E D is a light tail with associated minimal idempotent E 1 , by the definition of a light tail.The following result is a consequence of[1, Cor 3.4]. Let Γ be a distance-regular graph with valency k ≥ 3, diameter D ≥ 2 and distinct eigenvalues θ 0 > θ 1 > · · · > θ D .…”

Distance-regular graphs

“…This would be best possible as the Doubled Odd graphs have second largest eigenvalue k − 1. For distance-regular graphs with girth 6, it was shown by Bang, Koolen, and Park [35].…”

“…The quantum decomposition 34 of A is the expression A = L+F +R. The primary T-module, together with L and R, naturally has the structure of a one-mode interacting Fock space, 35 and they took the limit of the coefficients of the three-term recurrence relation of the associated orthogonal polynomials (which are certain normalizations of the v i from (2)) to get the quantum central limit theorem. See [335,336] for more details.…”

“…36 Γ 1 ∪ Γ 4 would be a strongly regular graph with parameters (324, 57, 0, 12), which does not exist [243]. 37 The halved graphs would be the complement of a strongly regular graph with parameters (456,35,10,2), which does not exist [92] by Proposition 9.5. 38 The halved graphs would be the complement of a strongly regular graph with parameters (726, 29, 4, 1), which does not exist by Proposition 9.5.…”

“…The quantum decomposition for the complete graph K 2 is sometimes referred to as a quantum cointossing 35. In this context, L and R are called the annihilation and the creation operators, respectively.…”