Coherent configurations, association schemes and permutation groups

Cameron, Peter J‎.

doi:10.1142/9789812564481_0004

Cited by 22 publications

(14 citation statements)

References 26 publications

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“…We use the same definition as the books of Bailey [7] and Brouwer, Cohen and Neumaier [22]; however, the books by Bannai and Ito [12] and Zieschang [96] each give different definitions, which are both weaker than ours. We refer the reader to [34] for a discussion of the various objects that have been called 'association schemes'.…”

Section: Coherent Configurations and Association Schemesmentioning

confidence: 99%

Base size, metric dimension and other invariants of groups and graphs

Bailey¹,

Cameron²

2011

Bulletin of the London Mathematical Society

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222

297

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The base size of a permutation group, and the metric dimension of a graph, are two of a number of related parameters of groups, graphs, coherent configurations and association schemes. They have been repeatedly redefined with different terminology in various different areas, including computational group theory and the graph isomorphism problem. We survey results on these parameters in their many incarnations, and propose a consistent terminology for them. We also present some new results, including on the base sizes of wreath products in the product action, and on the metric dimension of Johnson and Kneser graphs.

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Section: Coherent Configurations and Association Schemesmentioning

confidence: 99%

Base size, metric dimension and other invariants of groups and graphs

Bailey¹,

Cameron²

2011

Bulletin of the London Mathematical Society

Self Cite

222

297

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“…Association schemes and coherent configurations ( [1,6,19,20]), and the related Bose-Mesner algebras and coherent algebras ( [7]) are well-known in combinatorics with many applications in coding theory, graph theory and statistics (e.g., [2][3][4][5]8,9,11,12,16,17,24,28,33,34]).…”

Section: Association Schemes On Triplesmentioning

confidence: 99%

Circulant association schemes on triples

Bhattacharya¹,

Praeger²

2021

Preprint

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Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an association scheme on triples (AST) and constructed examples of several families of ASTs. Many of their examples used 2transitive permutation groups: the non-trivial ternary relations of the ASTs were sets of ordered triples of pairwise distinct points of the underlying set left invariant by the group; and the given permutation group was a subgroup of automorphisms of the AST. In this paper, we consider ASTs that do not necessarily admit 2-transitive groups as automorphism groups but instead a transitive cyclic subgroup of the symmetric group acts as automorphisms. Such ASTs are called circulant ASTs and the corresponding ternary relations are called circulant relations. We give a complete characterisation of circulant ASTs in terms of AST-regular partitions of the underlying set. We also show that a special type of circulant, that we call a thin circulant, plays a key role in describing the structure of circulant ASTs. We outline several open questions.Funding 1 .

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“…We introduce the Bose-Mesner algebra and relevant projections and applications here. For a more thorough discussion on Bose-Mesner algebras and their application in statistical and quantum physics, combinatorics, coding, graph theory, and statistical covariance analysis (more particularly in experimental designs) we refer to the seminal article of Bose-Mesner [20], the ones of Nelder [67,68], the more recent articles [32,24,40], as well as the books [21,10].…”

Section: Bose-mesner Projectionsmentioning

confidence: 99%

Perturbations and projections of Kalman–Bucy semigroups

Bishop

Moral

Pathiraja

2018

Stochastic Processes and their Applications

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We analyse various perturbations and projections of Kalman-Bucy semigroups and Riccati equations. For example, covariance inflation-type perturbations and localisation methods (projections) are common in the ensemble Kalman filtering literature. In the limit of these ensemble methods, the regularised sample covariance tends toward a solution of a perturbed/projected Riccati equation. With this motivation, results are given characterising the error between the nominal and regularised Riccati flows and Kalman-Bucy filtering distributions. New projectiontype models are also discussed; e.g. Bose-Mesner projections. These regularisation models are also of interest on their own, and in, e.g., differential games, control of stochastic/jump processes, and robust control.

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Coherent configurations, association schemes and permutation groups

Cited by 22 publications

References 26 publications

Base size, metric dimension and other invariants of groups and graphs

Base size, metric dimension and other invariants of groups and graphs

Circulant association schemes on triples

Perturbations and projections of Kalman–Bucy semigroups

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