Siamese Combinatorial Objects via Computer Algebra Experimentation

Klin, Mikhail; Reichard, Sven; Woldar, Andrew J.

doi:10.1007/978-3-642-01960-9_2

Cited by 7 publications

(11 citation statements)

References 36 publications

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“…Many texts, which were written at different times in diverse styles, might be quite helpful. Here is a small sample: [Bai04], [BI84], [Cam99], [KG15], [KPR + 10], [KRW09]. A number of other significant texts will be mentioned later on in an ad hoc manner.…”

Section: Preliminaries From Agt and Computer Algebramentioning

confidence: 99%

Proper Jordan schemes exist. First examples, computer search, patterns of reasoning. An essay

Klin,

Muzychuk,

Reichard

2019

Preprint

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A special class of Jordan algebras over a field F of characteristic zero is considered. Such an algebra consists of an r-dimensional subspace of the vector space of all square matrices of a fixed order n over F . It contains the identity matrix, the all-one matrix; it is closed with respect to matrix transposition, Schur-Hadamard (entrywise) multiplication and the Jordan product A * B = 1 2 (AB + BA), where AB is the usual matrix product. The suggested axiomatics (with some natural additional requirements) implies an equivalent reformulation in terms of symmetric binary relations on a vertex set of cardinality n. The appearing graph-theoretical structure is called a Jordan scheme of order n and rank r. A significant source of Jordan schemes stems from the symmetrization of association schemes. Each such structure is called a non-proper Jordan scheme. The question about the existence of proper Jordan schemes was posed a few times by Peter J. Cameron.In the current text an affirmative answer to this question is given. The first small examples presented here have orders n = 15, 24, 40. Infinite classes of proper Jordan schemes of rank 5 and larger are introduced. A prolific construction for schemes of rank 5 and order n =The text is written in the style of an essay. The long exposition relies on initial computer experiments, a large amount of diagrams, and finally is supported by a number of patterns of general theoretical reasonings. The essay contains also a historical survey and an extensive bibliography.

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Section: Preliminaries From Agt and Computer Algebramentioning

confidence: 99%

Proper Jordan schemes exist. First examples, computer search, patterns of reasoning. An essay

Klin,

Muzychuk,

Reichard

2019

Preprint

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“…Thus we here cite two of our papers [12,13] in which we provide various details of AGT in accordance with our own personal bias.…”

Section: More About Algebraic Graph Theorymentioning

confidence: 99%

“…The answer is exactly one. For the authors MK and AW, this result holds special significance, see [13].…”

Section: Interests Of Pickert In Design Theory That Touch Agtmentioning

confidence: 99%

On the existence of self-complementary and non-self-complementary strongly regular graphs with Paley parameters

2015

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For p an odd prime, let Ap be the complete classical affine association scheme whose associate classes correspond to parallel classes of lines in the classical affine plane AG(2, p). It is known that Ap is an amorphic association scheme. We investigate rank 3 fusion schemes of Ap whose basis graphs have the same parameters as the Paley graphs P (p 2 ). In contrast to the Paley graphs, the great majority of graphs we detect are non-self-complementary and non-Schurian. In particular, existence of non-self-complementary graphs with Paley parameters is established for p ≥ 17, with an analogous existence result for non-Schurian such graphs when p ≥ 11. We demonstrate that the number of self-complementary and non-self-complementary strongly regular graphs with Paley parameters grows rapidly as p → ∞.

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“…The distinction between explanation and interpretation can be both subtle and subjective, cf. [50]. Typically the word explanation is used.…”

Section: From Computer Experiments To Computer-free Interpretationmentioning

confidence: 99%

Examples of computer experimentation in algebraic combinatorics

Klin

Pech²,

Reichard³

et al. 2010

Ars Math. Contemp.

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We introduce certain paradigms for procuring computer-free explanations from data acquired via computer algebra experimentation. Our established context is the field of algebraic combinatorics, with special focus on coherent configurations and association schemes. All results presented here were obtained by the authors with the aid of computer algebra systems, especially COCO and GAP. A number of examples are explored, in particular of objects on 28, 50, 63, and 210 points. In a few cases, initial experimental data pointed to appropriate theoretical generalizations that yielded an infinite class of related combinatorial structures. Special attention is paid to algebraic automorphisms (of a coherent algebra), a fairly new concept that has already proved to have far-reaching consequences. Finally, we focus on the Doyle-Holt graph on 27 vertices, and some of its related structures.

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Siamese Combinatorial Objects via Computer Algebra Experimentation

Cited by 7 publications

References 36 publications

Proper Jordan schemes exist. First examples, computer search, patterns of reasoning. An essay

Proper Jordan schemes exist. First examples, computer search, patterns of reasoning. An essay

On the existence of self-complementary and non-self-complementary strongly regular graphs with Paley parameters

Examples of computer experimentation in algebraic combinatorics

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