A non-Schurian coherent configuration on 14 points exists

Klin, Mikhail; Ziv-Av, Matan

doi:10.1007/s10623-016-0258-8

Cited by 5 publications

(7 citation statements)

References 16 publications

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“…for the basis relations R A and R B introduced by (25). Based on (26), we conclude that the restriction of f to C[A ∪ B ∪ X] is induced by φ, as desired.…”

Section: Consider the Basis Relationsmentioning

confidence: 67%

“…This ensures that φ(R A ) = f (R A ) for the basis relations R A given by (25). Based on (26), we see that…”

Section: Case 1: Bothmentioning

confidence: 96%

“…Note that all coherent configuration with at most 15 points have been enumerated [26], but their separability analysis seems to be still missing. 2 Definability of a graph in 3-variable logic.…”

Section: Introductionmentioning

confidence: 99%

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Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm

Fuhlbrück¹,

Köbler²,

Verbitsky³

2019

Preprint

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As it is well known, the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-Fürer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2-WL, that is, whether or not 2-WL distinguishes G from any non-isomorphic graph. In fact, we solve the much more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to graphs of color multiplicity 4 with directed and colored edges.Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-Fürer-Immerman graphs of color multiplicity 4 appear here as a natural subclass, which demonstrates that the Cai-Fürer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as (n 3 )-configurations in incidence geometry.

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“…for the basis relations R A and R B introduced by (25). Based on (26), we conclude that the restriction of f to C[A ∪ B ∪ X] is induced by φ, as desired.…”

Section: Consider the Basis Relationsmentioning

confidence: 67%

“…This ensures that φ(R A ) = f (R A ) for the basis relations R A given by (25). Based on (26), we see that…”

Section: Case 1: Bothmentioning

confidence: 96%

See 1 more Smart Citation

Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm

Fuhlbrück¹,

Köbler²,

Verbitsky³

2019

Preprint

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show abstract

“…• CCs have been classified using a computer up to order 15 by Ziv-Av [38]. In this paper Klin and Ziv-Av also describe the smallest CC, with order 14, that is not a two-orbit configuration.…”

Section: The Algebrasmentioning

confidence: 99%

A survey of semisimple algebras in algebraic combinatorics

Herman

2021

Indian J Pure Appl Math

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This is a survey of semisimple algebras of current interest in algebraic combinatorics, with a focus on questions which we feel will be new and interesting to experts in group algebras, integral representation theory, and computational algebra. The algebras arise primarily in two families: coherent algebras and subconstituent (aka. Terwilliger) algebras. Coherent algebras are subalgebras of full matrix algebras having a basis of 01-matrices satisfying the conditions that it be transpose-closed, sum to the all 1's matrix, and contain a subset ∆ that sums to the identity matrix. The special case when ∆ is a singleton is the important case of an adjacency algebra of a finite association scheme. A Terwilliger algebra is a semisimple extension of a coherent algebra by a set of diagonal 01-matrices determined canonically from its basis elements and a choice of row. We will survey the current state of knowledge of the complex, real, rational, modular, and integral representation theory of these semisimple algebras, indicate their connections with other areas of mathematics, and present several open questions.

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“…Each such CC is called Schurian CC, otherwise it is called non-Schurian. According to [KZA17] all CCs of order n ≤ 13 are Schurian, while there exists an example of a non-Schurian CC of order 14. Similar wording is used for the particular case of ASs.…”

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

Self Cite

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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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A non-Schurian coherent configuration on 14 points exists

Cited by 5 publications

References 16 publications

Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm

Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm

A survey of semisimple algebras in algebraic combinatorics

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

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