On the Terwilliger algebra of bipartite distance-regular graphs with [Delta][sub]2 = 0 and c[sub]2=1

MacLean, Mark; Miklavič, Štefko; Penjić, Safet

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Title:	On the Terwilliger algebra of bipartite distance-regular graphs with [Delta][sub]2 = 0 and c[sub]2=1
Authors:	ID MacLean, Mark (Author) ID Miklavič, Štefko (Author) ID Penjić, Safet (Author)
Files:	http://dx.doi.org/10.1016/j.laa.2016.01.040
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
Abstract:	Let $\Gamma$ denote a bipartite distance-regular graph with diameter $D \geq 4$ and valency $k \geq 3$ . Let $X$ denote the vertex set of $\Gamma$ , and let $A$ denote the adjacency matrix of $\Gamma$ . For $x \in X$ and for $0 \leq i \leq D$ , let $\operatorname{\Gamma}_i(x)$ denote the set of vertices in $X$ that are distance $i$ from vertex $x$ . Define a parameter $\operatorname{\Delta}_2$ in terms of the intersection numbers by $\operatorname{\Delta}_2 = (k - 2)(c_3 - 1) -(c_2 - 1) p_{22}^2$ . We first show that $\operatorname{\Delta}_2 = 0$ implies that $D \leq 5$ or $c_2 \in \{1, 2 \}$ . For $x \in X$ let $T = T(x)$ denote the subalgebra of $\text{Mat}_X(\mathbb{C})$ generated by $A, E_0^\ast, E_1^\ast, \ldots, E_D^\ast$ , where for $0 \leq i \leq D$, $E_i^\ast$ represents the projection onto the▫ $i$ ▫th subconstituent of $\Gamma$ with respect to $x$ . We refer to $T$ as the Terwilliger algebra of $\Gamma$ with respect to $x$ . By the endpoint of an irreducible $T$ -module $W$ we mean $\min \{i \| E_i^\ast W \ne 0 \}$ . In this paper we assume $\Gamma$ has the property that for $2 \leq i \leq D - 1$ , there exist complex scalars $\alpha_i$ , $\beta_i$ such that for all $x, y, z \in X$ with $\partial(x, y) = 2$ , $\partial(x, z) = i$ , $\partial(y, z) = i$ , we have $\alpha_i + \beta_i \| \operatorname{\Gamma}_1(x) \cap \operatorname{\Gamma}_1(y) \cap \operatorname{\Gamma}_{i - 1}(z) \| = \| \operatorname{\Gamma}_{i - 1}(x) \cap \operatorname{\Gamma}_{i - 1}(y) \cap \operatorname{\Gamma}_1(z) \|$ . We additionally assume that▫ $\operatorname{\Delta}_2 = 0$ ▫ with $c_2 = 1$ . Under the above assumptions we study the algebra $T$ . We show that if $\Gamma$ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible $T$ -module with endpoint 2. We give an orthogonal basis for this $T$ -module, and we give the action of $A$ on this basis.
Keywords:	distance-regular graphs, terwilliger algebra, subconstituent algebra
Year of publishing:	2016
Number of pages:	str. 307-330
Numbering:	Vol. 496
PID:	20.500.12556/RUP-8840
ISSN:	0024-3795
UDC:	519.17
DOI:	10.1016/j.laa.2016.01.040
COBISS.SI-ID:	1538163396
Publication date in RUP:	14.11.2017
Views:	3134
Downloads:	146
Metadata:
:	MACLEAN, Mark, MIKLAVIČ, Štefko and PENJIĆ, Safet, 2016, On the Terwilliger algebra of bipartite distance-regular graphs with [Delta][sub]2 = 0 and c[sub]2=1. [online]. 2016. Vol. 496, p. 307–330. [Accessed 5 April 2025]. DOI 10.1016/j.laa.2016.01.040. Retrieved from: http://dx.doi.org/10.1016/j.laa.2016.01.040 Copy citation

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