Weighted association schemes, fusions, and minimal coherent closures

Sankey, A. D.

doi:10.1007/s10801-014-0553-2

Cited by 4 publications

(4 citation statements)

References 19 publications

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“…Noting that K 2 = −I it is straight-forward to see that Span(A ω k ) is coherent. For regularity, we note that p k ij is nonzero only when i + j + k is even, and this implies β k ij (±i) = 0 for all i, j, k. Proposition 1 of [21] applies, and we conclude that ω is regular.…”

Section: Necessary Conditions For a Covering Configurationmentioning

confidence: 66%

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On t-fold covers of coherent configurations

Sankey

2017

Ars Math. Contemp.

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We introduce the covering configuration induced by a regular weight defined on a coherent configuration. This construction generalizes the well-known equivalence of regular two-graphs and antipodal double covers of complete graphs. It also recovers, as special cases, the rank 6 association schemes connected with regular 3-graphs, and certain extended Q-bipartite doubles of cometric association schemes. We articulate sufficient conditions on the parameters of a coherent configuration for it to arise as a covering configuration.

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Section: Necessary Conditions For a Covering Configurationmentioning

confidence: 66%

“…has minimal closure if the fission {A α i } forms a CC. The terminology draws on the notion of the coherent closure of a set of matrices as the smallest CA containing them (see [21,28] for more). The coherent closure of (A, ω) is the CC whose CA is the coherent closure of the matrix algebra A ω .…”

Section: The Fission Induced By a Weightmentioning

confidence: 99%

On t-fold covers of coherent configurations

Sankey

2017

Ars Math. Contemp.

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“…A good introduction to the topic may be found in [41]. In the literature, a rich theory has been built up around this concept, and much more can be found in [38,39,42,60,61,62,63,72]. It is well known that every coherent algebra C is semisimple (see, for example, [31, Section 2]) and that has a standard basis {N 0 , N 1 , .…”

Section: Introductionmentioning

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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“…A good introduction to the topic may be found in [41]. In the literature, a rich theory has been built up around this concept, and much more can be found in [37,38,40,59,60,61,62,70]. It is well known that every coherent algebra C is semisimple (see, for example, [31, Section 2]) and that has a standard basis {N 0 , N 1 , .…”

Section: Introductionmentioning

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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Weighted association schemes, fusions, and minimal coherent closures

Cited by 4 publications

References 19 publications

On t-fold covers of coherent configurations

On t-fold covers of coherent configurations

On symmetric association schemes and associated quotient-polynomial graphs

On symmetric association schemes and associated quotient-polynomial graphs

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