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

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;mathematics;grah theory;distance-regular graphs;▫$Q$▫-polynomial property;equitable partitions;

Data

Language:	English
Year of publishing:	2007
Typology:	1.01 - Original Scientific Article
Organization:	UP - University of Primorska
UDC:	519.17
COBISS:	14181465
ISSN:	0012-365X
Views:	3875
Downloads:	37
Average score:	0 (0 votes)
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Other data

Secondary language:	Slovenian
Secondary title:	O dvodelnih Q-polinomskih razdaljno regularnih grafih s presečnim številom 1
Secondary 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 ▫$d=4$▫.
Secondary keywords:	matematika;teorija grafov;razdaljno regularni grafi;▫$Q$▫-polinomska lastnost;ekvitabilne particije;
Type (COBISS):	Not categorized
Pages:	str. 544-553
Volume:	ǂVol. ǂ307
Issue:	ǂiss. ǂ3-5
Chronology:	2007
DOI:	10.1016/j.disc.2005.09.044
ID:	1472912