Light tails and the Hermitian dual polar graphs

Jurisic et al. (Eur J Comb 31:1539-1552, 2010) conjectured (see also [van Dam et al. in Distance-regular graphs, Problem 13]) that if a distance-regular graph with diameter D at least three has a light tail, then one of the following holds: (1) a(1) = 0; (2) Gamma is an antipodal cover of diameter three; (3) Gamma is tight; (4) Gamma is the halved (2D+1)-cube; and (5) Gamma is a Hermitian dual polar graph(2)A(2D-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/a(1)+1. Our first main result is: Theorem Let Gamma be a non-bipartite distance-regular graph with valency k >= 3, diameter D >= 3 and distinct eigenvalues theta(0)>theta(1)>...>theta(D). Suppose that Gamma is 2-bounded with smallest eigenvalue theta(D) = -k/a(1)+1. If the minimal idempotent E-D corresponding to eigenvalue theta(D) is a light tail, then Gamma is the dual polar graph(2) A(2D-1)(r), where r=a(1)+1 is a prime power. As a consequence of this result we will show our second main result: Theorem Let Gamma be a distance-regular graph with valency k >= 3, diameter D >= 2, and a(1)=1 and theta(0)>theta(1)>...>theta(D). If c(2)>= 4 and theta(D)=-k/2, then c(2)=5 and Gamma is the dual polar graph(2)A(2D-1)(2) .