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

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Title:	On bipartite Q-polynomial distance-regular graphs with c [sub] 2 [equal] 1
Authors:	ID Miklavič, Štefko (Author)
Files:	http://dx.doi.org/10.1016/j.disc.2005.09.044
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
Abstract:	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 ▫$d=4$▫.
Keywords:	mathematics, grah theory, distance-regular graphs, ▫$Q$▫-polynomial property, equitable partitions
Year of publishing:	2007
Number of pages:	str. 544-553
Numbering:	Vol. 307, iss. 3-5
PID:	20.500.12556/RUP-286
ISSN:	0012-365X
UDC:	519.17
COBISS.SI-ID:	14181465
Publication date in RUP:	15.10.2013
Views:	4644
Downloads:	38
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Language:	Slovenian
Abstract:	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 ▫$d=4$▫.
Keywords:	matematika, teorija grafov, razdaljno regularni grafi, ▫$Q$▫-polinomska lastnost, ekvitabilne particije

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