On bipartite Q-polynominal distance-regular graphs

Miklavič, Štefko

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Title:	On bipartite Q-polynominal distance-regular graphs
Authors:	ID Miklavič, Štefko (Author)
Files:	http://dx.doi.org/10.1016/j.ejc.2005.09.003
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 vertex set $X$ , diameter $d \ge 3$ and valency $k \ge 3$ . Let ${\mathbb{R}}^X$ denote the vector space over $\mathbb{R}$ consisting of column vectors with entries in $\mathbb{r}$ and rows indexed by $X$ . For $z \in X$ , let $\hat{z}$ denote the vector in ${\mathbb{R}}^X$ with a 1 in the $z$ -coordinate, and 0 in all other coordinates. Fix $x,y \in X$ such that $\partial(x,y)=2▫, where ▫$\partial$ denotes the path-length distance. For $0 \le i,j \le d$ define $w_{ij} = \sum\hat{z}$ , where the sum is over all $z \in X$ such that $\partial(x,z) = i$ and $\partial(y,z) = j▫$. We define ▫$W = \textrm{span} \{w_{ij}\|0 \le i,j \le d\}$ . In this paper we consider the space $MW = \textrm{span} \{mw \|m \in M, w \in W \l\}$ , where $M$ is the Bose-Mesner algebra of $\Gamma$ . We observe that $MW$ is the minimal $A$ -invariant subspace of ${\mathbb{R}}^X$ which contains $W$ , where $A$ is the adjacency matrix of $\Gamma$ . We display a basis for $MW$ that is orthogonal with respect to the dot product. We give the action of $A$ on this basis. We show that the dimension of $MW$ is $3d-3$ if $\Gamma$ is 2-homogeneous, $3d-1$ if $\Gamma$ is the antipodal quotient of the $2d$ -cube, and $4d-4$ otherwise. We obtain our main result using Terwilliger's "balanced set" characterization of the $Q$ -polynomial property.
Keywords:	mathematics, graph theory, distance-regular graphs, $Q$ -polynominal property, Bose-Mesner algebra, balanced set characterization of the Q-polynominal property
Year of publishing:	2007
Number of pages:	str. 94-110
Numbering:	Vol. 28, no. 1
PID:	20.500.12556/RUP-3312
ISSN:	0195-6698
UDC:	519.17
COBISS.SI-ID:	1796823
Publication date in RUP:	15.10.2013
Views:	5400
Downloads:	30
Metadata:
:	MIKLAVIČ, Štefko, 2007, On bipartite Q-polynominal distance-regular graphs. [online]. 2007. Vol. 28, no. 1, p. 94–110. [Accessed 6 April 2025]. Retrieved from: http://dx.doi.org/10.1016/j.ejc.2005.09.003 Copy citation