On bipartite Q-polynomial distance-regular graphs with c [sub] 2 [equal] 1

Let ▫$Gamma$▫ denote a bipartite ▫$Q$▫-polynomial distance-regular graph with diameter ▫$d ge 3$▫, valency ▫$k ge 3$▫ and intersection number ▫$c_2=1$▫. We show that ▫$Gamma$▫ has a certain equitable partition of its vertex set which involves ▫$4d-4$▫ cells. We use this partition to show that the intersection numbers of ▫$Gamma$▫ satisfy the following divisibility conditions: (I) ▫$c_{i+1}-1$▫ divides ▫$c_i(c_i-1)$▫ for ▫$2 le i le d-1$▫, and (II) ▫$b_{i-1}-1$▫ divides ▫$b_i(b_i-1)$▫ for ▫$1 le i le d-1$▫. Using these divisibility conditions we show that ▫$Gamma$▫ does not exist if ▫$d=4$▫.Naj bo ▫$Gamma$▫ dvodelen ▫$Q$▫-polinomski razdaljno regularen graf premera ▫$d ge 3$▫, stopnje ▫$k ge 3$▫ in presečnim številom ▫$c_2=1$▫. Pokažemo, da množica vozlišč grafa ▫$Gamma$▫ premore ekvitabilno particijo, ki vsebuje ▫$4d-4$▫ množic. S pomočjo te ekvitabilne particije dokaemo, da morajo presečna števila grafa ▫$Gamma$▫ zadoščati naslednjim pogojem: (I) ▫$c_{i+1}-1$▫ deli ▫$c_i(c_i-1)$▫ za ▫$2 le i le d-1$▫, (II) ▫$b_{i-1}-1$▫ deli ▫$b_i(b_i-1)$▫ za ▫$1 le i le d-1$▫. S pomočjo teh pogojev dokažemo, da graf ▫$Gamma$▫ ne obstaja, če je ▫$d=4$▫