Small latin squares, quasigroups, and loops

McKay, Brendan D.; Meynert, Alison; Myrvold, Wendy

doi:10.1002/jcd.20105

Cited by 133 publications

(163 citation statements)

References 35 publications

(40 reference statements)

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“…i k i !. We note here that a similar result was derived by mathematicians for the case in which the maximal connectivity K < N 1/3 [37] and an inequality was proved for the case K > N 1/3 [38]. Now we extend these results by statistical mechanics methods to uncorrelated networks with maximal connectivity K < k N .…”

Section: Uncorrelated Networksupporting

confidence: 78%

Entropy of network ensembles

2009

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In this paper we generalize the concept of random networks to describe networks with non trivial features by a statistical mechanics approach. This framework is able to describe ensembles of undirected, directed as well as weighted networks. These networks might have not trivial community structure or, in the case of networks embedded in a given space, non trivial distance dependence of the link probability. These ensembles are characterized by their entropy which evaluate the cardinality of networks in the ensemble. The general framework we present in this paper is able to describe microcanonical ensemble of networks as well as canonical or hidden variables network ensemble with significant implication for the formulation of network constructing algorithms. Moreover in the paper we define and and characterize in particular the structural entropy, i.e. the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence. We discuss the apparent paradox that scale-free degree distribution are characterized by having small structural entropy but are so widely encountered in natural, social and technological complex systems.We give the proof that while scale-free networks ensembles have small structural entropy, they also correspond to the most likely degree distribution with the corresponding value of the structural entropy.

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Section: Uncorrelated Networksupporting

confidence: 78%

Entropy of network ensembles

2009

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“…The enumeration of latin squares has a lengthy and well known history for which we refer to the recent survey given in [18]. Considerably less work has been done for higher dimensions.…”

Section: Basic Definitionsmentioning

confidence: 99%

“…The procedure for counting isomorphism classes of ordinary quasigroups given in [18,Theorem 4] applies equally well to the d-ary case, and the same proof applies with obvious adaptions, so we state our first theorem without proof.…”

Section: Counting Equivalence Classesmentioning

confidence: 99%

A Census of Small Latin Hypercubes

McKay¹,

Wanless²

2008

SIAM J. Discrete Math.

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We count all latin cubes of order n ≤ 6 and latin hypercubes of order n ≤ 5 and dimension d ≤ 5. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of n and d we classify all d-ary quasigroups of order n into isomorphism classes and also count them according to the number of identity elements they possess (meaning we have counted the d-ary loops).We also give an exact formula for the number of (isomorphism classes of) d-ary quasigroups of order 3 for every d. Then we give a number of constructions for d-ary quasigroups with a specific number of identity elements. In the process, we prove that no 3-ary loop of order n can have exactly n − 1 identity elements (but no such result holds in dimensions other than 3). Finally, we give some new examples of latin cuboids which cannot be extended to latin cubes.

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“…Recently, McKay, Meynert, and Myrvold [14] successfully applied this general idea to the problem of enumerating Latin squares. In the present paper, it is applied to the problem of enumerating one-factorizations of the complete graph K 14 .…”

Section: Introductionmentioning

confidence: 99%