On Lee association schemes over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:math> and their Terwilliger algebra

Morales, John Vincent S.

doi:10.1016/j.laa.2016.08.033

Cited by 10 publications

(3 citation statements)

References 18 publications

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“…The third equality holds by Lemma 5.18. The fourth equality holds by (5), (16), and Definition 5.11. The last equality follows by Lemma 6.2 and since E * r+j z is a common eigenvector for M * .…”

Section: Irreducible T -Modules and Q-modulesmentioning

confidence: 97%

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The quantum adjacency algebra and subconstituent algebra of a graph

Terwilliger

Žitnik

2019

Journal of Combinatorial Theory, Series A

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Let Γ denote a finite, undirected, connected graph, with vertex set X. Fix a vertex x ∈ X. Associated with x is a certain subalgebra T = T (x) of Mat X (C), called the subconstituent algebra. The algebra T is semisimple. Hora and Obata introduced a certain subalgebra Q ⊆ T , called the quantum adjacency algebra. The algebra Q is semisimple. In this paper we investigate how Q and T are related. In many cases Q = T , but this is not true in general. To clarify this issue, we introduce the notion of quasi-isomorphic irreducible T -modules. We show that the following are equivalent: (i) Q = T ; (ii) there exists a pair of quasi-isomorphic irreducible T -modules that have different endpoints. To illustrate this result we consider two examples. The first example concerns the Hamming graphs. The second example concerns the bipartite dual polar graphs. We show that for the first example Q = T , and for the second example Q = T .

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Section: Irreducible T -Modules and Q-modulesmentioning

confidence: 97%

“…In [20] the subconstituent algebra was introduced, and used to investigate commutative association schemes. Since then the subconstituent algebra has received considerable attention; some notable papers are [3,5,7,8,10,11,12,13,15,16,17,18,19,23,25].…”

Section: Introductionmentioning

confidence: 99%

The quantum adjacency algebra and subconstituent algebra of a graph

Terwilliger

Žitnik

2019

Journal of Combinatorial Theory, Series A

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“…This T -module concept is also taken from the theory of commutative association schemes [63,64,65]. For most recent research on the use of Terwilliger algebra in the study of P -polynomial association schemes (that is, using the Terwilliger algebra to study distance-regular graphs) see [11,44,45,47,46,51,52,53,58].…”

Section: Further Directionsmentioning

confidence: 99%

On symmetric association schemes and associated quotient-polynomial graphs

Fiol¹,

Penjić²

2020

Preprint

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Let Γ denote an undirected, connected, regular graph with vertex set X, adjacency matrix A, and d + 1 distinct eigenvalues. Let A = A(Γ) denote the subalgebra of Mat X (C) generated by A. We refer to A as the adjacency algebra of Γ. In this paper we investigate algebraic and combinatorial structure of Γ for which the adjacency algebra A is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) A has a standard basis {I, F 1 , . . . , F d }; (ii) for every vertex there exists identical distance-faithful intersection diagram of Γ with d + 1 cells; (iii) the graph Γ is quotient-polynomial; and (iv) if we pick F ∈ {I, F 1 , . . . , F d } then F has d + 1 distinct eigenvalues if and only if span{I, F 1 , . . . , F d } = span{I, F, . . . , F d }. We describe the combinatorial structure of quotient-polynomial graphs with diameter 2 and 4 distinct eigenvalues. As a consequence of the technique from the paper we give an algorithm which computes the number of distinct eigenvalues of any Hermitian matrix using only elementary operations. When such a matrix is the adjacency matrix of a graph Γ, a simple variation of the algorithm allow us to decide wheter Γ is distance-regular or not. In this context, we also propose an algorithm to find which distance-i matrices are polynomial in A, giving also these polynomials.

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An Assmus–Mattson theorem for codes over commutative association schemes

Morales

Tanaka

2017

Des. Codes Cryptogr.

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On Lee association schemes over Z4 and their Terwilliger algebra

Cited by 10 publications

References 18 publications

The quantum adjacency algebra and subconstituent algebra of a graph

The quantum adjacency algebra and subconstituent algebra of a graph

On symmetric association schemes and associated quotient-polynomial graphs

An Assmus–Mattson theorem for codes over commutative association schemes

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