Algebraic decompositions of commutative association schemes

Ferguson, Pamela A.; Turull, Alexandre

doi:10.1016/0021-8693(85)90047-x

Cited by 14 publications

(3 citation statements)

References 6 publications

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“…The Bose-Mesner algebra was introduced in [7], and the monumental thesis of Delsarte [17] proclaimed the importance of commutative association schemes as a unifying framework for coding theory and design theory. There are a number of excellent articles and textbooks on the theory of (commutative) association schemes and Delsarte's theory; see, for instance, [4,8,18,21,40,51]. The following are some of the books which include accounts on commutative association schemes: [10,32,49,44].…”

Section: (As2)mentioning

confidence: 99%

On symmetric association schemes and associated quotient-polynomial graphs

Fiol¹,

Penjić²

2022

Algebraic Combinatorics

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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 distancefaithful 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 quotientpolynomial graphs with diameter 2 and 4 distinct eigenvalues. As a consequence of the techniques used in the paper, some simple algorithms allow us to decide whether Γ is distance-regular or not and, more generally, which distance-i matrices are polynomial in A, giving also these polynomials.

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Section: (As2)mentioning

confidence: 99%

On symmetric association schemes and associated quotient-polynomial graphs

Fiol¹,

Penjić²

2022

Algebraic Combinatorics

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“…The Bose-Mesner algebra was introduced in [7], and the monumental thesis of Delsarte [17] proclaimed the importance of commutative association schemes as a unifying framework for coding theory and design theory. There are a number of excellent articles and textbooks on the theory of (commutative) association schemes and Delsarte's theory; see, for instance, [4,8,18,22,39,50]. The following are some of the books which include accounts on commutative association schemes: [10,32,48,43].…”

Section: (As2)mentioning

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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“…A crucial contribution, exploited in [22,23], was the use of the group basis in the complete rational S-ring over H. Implicitly or explicitly the algebras A n and V (H, Z * n ) were investigated later on again and again, basing on diverse motivation see e.g. [101,43,45,15].…”

Section: Historical Digestmentioning

confidence: 99%

Automorphism Groups of Rational Circulant Graphs

Klin¹,

Kovács²

2012

Electron. J. Combin.

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The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups.

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Algebraic decompositions of commutative association schemes

Cited by 14 publications

References 6 publications

On symmetric association schemes and associated quotient-polynomial graphs

On symmetric association schemes and associated quotient-polynomial graphs

On symmetric association schemes and associated quotient-polynomial graphs

Automorphism Groups of Rational Circulant Graphs

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