Q-polynomial distance-regular graphs with a [sub] 1 [equal] 0 and a [sub] 2 [not equal] 0

Miklavič, Štefko

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Title:	Q-polynomial distance-regular graphs with a [sub] 1 [equal] 0 and a [sub] 2 [not equal] 0
Authors:	ID Miklavič, Štefko (Author)
Files:	http://dx.doi.org/10.1016/j.ejc.2008.02.001
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
Abstract:	Let ▫$\Gamma$▫ denote a ▫$Q$▫-polynomial distance-regular graph with diameter ▫$D \ge 3$▫ and intersection numbers ▫$a_1=0$▫, ▫$a_2 \ne 0$▫. Let ▫$X$▫ denote the vertex set of ▫$\Gamma$▫ and let ▫$A \in {\mathrm{Mat}}_X ({\mathbb{C}})$▫ denote the adjacency matrix of ▫$\Gamma$▫. Fix ▫$x \in X$▫ and let denote $A^\ast \in {\mathrm{Mat}}_X ({\mathbb{C}})$ the corresponding dual adjacency matrix. Let ▫$T$▫ denote the subalgebra of ▫$A{\mathrm{Mat}}_X ({\mathbb{C}})$▫ generated by ▫$A$▫, ▫$A^\ast$▫. We call ▫$T$▫ the Terwilliger algebra of ▫$\Gamma$▫ with respect to ▫$x$▫. We show that up to isomorphism there exists a unique irreducible ▫$T$▫-module ▫$W$▫ with endpoint 1. We show that ▫$W$▫ has dimension ▫$2D-2$▫. We display a basis for ▫$W$▫ which consists of eigenvectors for ▫$A^\ast$▫. We display the action of ▫$A$▫ on this basis. We show that ▫$W$▫ appears in the standard module of ▫$\Gamma$▫ with multiplicity ▫$k-1$▫, where ▫$k$▫ is the valency of ▫$\Gamma$▫.
Keywords:	mathematics, graph theory, adjacency matrix, distance-regular graph, Terwilliger algebra
Year of publishing:	2008
Number of pages:	str. 192-207
Numbering:	Vol. 30, no. 1
PID:	20.500.12556/RUP-1200
ISSN:	0195-6698
UDC:	519.17
COBISS.SI-ID:	14627929
Publication date in RUP:	15.10.2013
Views:	4396
Downloads:	31
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