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:	5227
Downloads:	39
Metadata:
:	MIKLAVIČ, Štefko, 2007, On bipartite Q-polynomial distance-regular graphs with c [sub] 2 [equal] 1. [online]. 2007. Vol. 307, no. 3–5, p. 544–553. [Accessed 1 April 2025]. Retrieved from: http://dx.doi.org/10.1016/j.disc.2005.09.044 Copy citation

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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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