Bipartite Distance-Regular Graphs with an Eigenvalue of Multiplicity \itk\rm

Yamazaki, Norio

doi:10.1006/jctb.1996.0004

Cited by 18 publications

(17 citation statements)

References 4 publications

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“…Since Y=Q 2 , we find that (7) by (12) and (19). K By [8], the 2-homogeneous bipartite distance-regular graphs are the only distance-regular graphs with a U q (sl(2)) structure like that of Theorem 3.5.…”

Section: And1mentioning

confidence: 94%

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The Terwilliger Algebra of a 2-Homogeneous Bipartite Distance-Regular Graph

Curtin¹

2001

Journal of Combinatorial Theory, Series B

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Let 1 denote a 2-homogeneous bipartite distance-regular graph with diameter D 3 and valency k 3. Assume that 1 is not isomorphic to a Hamming cube. Fix a vertex x of 1, and let T=T(x) denote the Terwilliger algebra of T with respect to x. We give three sets of generators for T, two of which satisfy the relations of the quantum universal enveloping algebra of the Lie algebra sl(2). We then describe the simple T-modules. We give a pair of canonical bases for each simple T-module, and we give the overlap function for these bases in terms of a basic hypergeometric function. Finally, we give two generators for the center of T. Academic Press

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Section: And1mentioning

confidence: 94%

“…It has the Q-polynomial property [6,16]. It is extremal in several algebraic and combinatorial properties [6,19]. By [8], it is related to U q (sl(2)), the quantum universal enveloping algebra of sl (2), and by [16] it is related to a spin model.…”

Section: Introductionmentioning

confidence: 99%

The Terwilliger Algebra of a 2-Homogeneous Bipartite Distance-Regular Graph

Curtin¹

2001

Journal of Combinatorial Theory, Series B

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“…The 2-homogeneous bipartite distance-regular graphs have been studied in [9,13,20,23]. In this paper we study a slightly more general class of graphs.…”

Section: Introductionmentioning

confidence: 99%

Almost 2-homogeneous Bipartite Distance-regular Graphs

Curtin

2000

European Journal of Combinatorics

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“…Let q ∈ {0, 1, −1} be any complex scalar such that (11), (12) hold, and let e i (0 ≤ i ≤ d) be as in (12). Then the matrices…”

Section: Lemma 35 Suppose Is Bipartite and 2-homogeneous But Not Isomentioning

confidence: 99%

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Curtin

Nomura

2000

Journal of Algebraic Combinatorics

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Abstract. We investigate a connection between distance-regular graphs and U q (sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let be a distance-regular graph with diameter d ≥ 3 and valency k ≥ 3, and assume is not isomorphic to the d-cube. Fix a vertex x of , and let T = T (x) denote the Terwilliger algebra of with respect to x. Fix any complex number q ∈ {0, 1, −1}. Then T is generated by certain matrices satisfying the defining relations of U q (sl(2)) if and only if is bipartite and 2-homogeneous.

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Bipartite Distance-Regular Graphs with an Eigenvalue of Multiplicity \itk\rm

Cited by 18 publications

References 4 publications

The Terwilliger Algebra of a 2-Homogeneous Bipartite Distance-Regular Graph

The Terwilliger Algebra of a 2-Homogeneous Bipartite Distance-Regular Graph

Almost 2-homogeneous Bipartite Distance-regular Graphs

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