Twice Q-Polynomial Distance-Regular Graphs

LetΓbe a distance-regular graph of diameter at least three. SupposeΓis Q-polynomial with respect to distinct eigenvaluesθandψofΓ. We calculate thekite numberf2j(x, y, z) from the intersection numbers ofΓand the dual eigenvalues toθandψ. This implies that the kite numbersfij(x, y, z) are independent of the verticesx,y,z. We use this result to show that the vertex neighborhood graphs ofΓare strongly regular and we calculate the parameters of these graphs from the kite numbersf2jand the intersection numbers ofΓ.