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:	6800
Downloads:	34
Metadata:
:	MIKLAVIČ, Štefko, 2008, Q-polynomial distance-regular graphs with a [sub] 1 [equal] 0 and a [sub] 2 [not equal] 0. [online]. 2008. Vol. 30, no. 1, p. 192–207. [Accessed 9 April 2025]. Retrieved from: http://dx.doi.org/10.1016/j.ejc.2008.02.001 Copy citation