20.500.12556/RUP-286 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 doka\emo, 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 ▫$d=4$▫. mathematics grah theory distance-regular graphs ▫$Q$▫-polynomial property equitable partitions matematika teorija grafov razdaljno regularni grafi ▫$Q$▫-polinomska lastnost ekvitabilne particije true true false Angleški jezik Slovenski jezik Delo ni kategorizirano 2013-10-15 12:04:40 2013-10-15 12:04:40 2024-03-01 11:55:03 0000-00-00 00:00:00 2007 0 0 str. 544-553 iss. 3-5 Vol. 307 2007 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 0012-365X 519.17 14181465 http://dx.doi.org/10.1016/j.disc.2005.09.044 1 https://repozitorij.upr.si/Dokument.php?lang=slv&id=286 Inštitut Andrej Marušič 0 0 0