Light tails and the Hermitian dual polar graphs

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Juriśič et al. conjectured that if a distance-regular graph Γ with diameter D at least three has a light tail, then one of the following holds: 1.a₁ =0; 2.Γ is an antipodal cover of diameter three; 3.Γ is tight; 4.Γ is the halved 2D+1-cube; 5.Γ is a Hermitian dual polar graph ²A₂D₋₁(r) where r is a prime power. In this note, we will consider the case when the light tail corresponds to the eigenvalue -k/a₁ +1. Our main result is: Theorem Let Γ be a non-bipartite distance-regular graph with valency k ≥ 3, diameter D ≥ 3 and distinct eigenvalues θ₀ > θ₁ > > θD. Suppose that Γ is 2-bounded with smallest eigenvalue θD = -k/a₁ +1. If the minimal idempotent ED, corresponding to eigenvalue θD, is a light tail, then Γ is the dual polar graph ²A₂D₋₁(r), where r is a prime power. As a consequence of this result we will also show: Theorem Let Γ be a distance-regular graph with valency k ≥ 3, diameter D ≥ 2, a₁ =1 and θ₀ > θ₁ > > θD. If c₂ ≥5 and θD = -k/2, then c₂ =5 and Γ is the dual polar graph ²A₂D₋₁(2).