Counting circulant graphs of prime-power order by decomposing into orbit enumeration problems

Liskovets, Valery A.; Pöschel, Reinhard

doi:10.1016/s0012-365x(99)00139-9

Cited by 14 publications

(17 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first attempt to enumerate Schur rings began with Liskovets and Pöschel [21] who enumerated wreath-indecomposable Schur rings over the cyclic group 2 We point out that all the cyclic group examples considered in Section 4 are Schurian by this theorem.…”

Section: The First Attempt To Classifymentioning

confidence: 99%

Enumeration Techniques on Cyclic Schur Rings

Misseldine

2021

Preprint

View full text Add to dashboard Cite

show abstract

Section: The First Attempt To Classifymentioning

confidence: 99%

Enumeration Techniques on Cyclic Schur Rings

Misseldine

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…We will see these again when we consider Schur rings over cyclic 2-groups in Section 6. In [13], Liskovets and Pöschel determine a formula for wreath-indecomposable Schur rings over Z p n , where p is an odd prime. This formula depends on the Catalan numbers and the number of divisors of p − 1.…”

Section: Introductionmentioning

confidence: 99%

Counting Schur rings over cyclic groups

Misseldine

2019

J Algebr Comb

View full text Add to dashboard Cite

Any Schur ring is uniquely determined by a partition of the elements of the group. An open question in the study of Schur rings is determining which partitions of the group induce a Schur ring. Although a structure theorem is available for Schur rings over cyclic groups, it is still a difficult problem to count all the partitions. For example, Kovacs, Liskovets, and Poschel determine formulas to count the number of wreath-indecomposable Schur rings. In this paper we solve the problem of counting the number of all Schur rings over cyclic groups of prime power order and draw some parallels with Higman's PORC conjecture.

show abstract

“…The analytic approach is based on the familiar isomorphism theorem [23] for circulant graphs of prime-power orders. In analytic enumeration we are guided also by the subsequent adaptation of this theorem to the enumeration of circulant graphs as developed in [32]. In particular, for circulant graphs of order p 3 their analytical (that is, formula-wise) enumeration has been reduced in this paper to five well-specified (and rather sophisticated) enumeration subproblems of Redfield-Pólya type.…”

Section: Introductionmentioning

confidence: 99%

Constructive and analytic enumeration of circulant graphs with $p^3$ vertices; $p=3,5$

Gatt,

Klin,

Lauri

et al. 2015

Preprint

Self Cite

View full text Add to dashboard Cite

Two methods, structural (constructive) and multiplier (analytical), of exact enumeration of undirected and directed circulant graphs of orders 27 and 125 are elaborated and represented in detail here together with intermediate and final numerical data. The first method is based on the known useful classification of circulant graphs in terms of S-rings and results in exhaustive listing (with the use of COCO and GAP) of all corresponding S-rings of the indicated orders. The latter method is conducted in the framework of a general approach developed earlier for counting circulant graphs of prime-power orders. It is a Redfield-Pólya type of enumeration based on an isomorphism criterion for circulant * Supported by Project Mobility Grant (see Acknowledgements) 1 graphs of such orders. In particular, five intermediate enumeration subproblems arise, which are refined further into eleven subproblems of this type (5 and 11 are, not accidentally, the 3d Catalan and 3d little Schröder numbers, resp.). All of them are resolved for the four cases under consideration (again with the use of GAP). We give a brief survey of some background theory of the results which form the basis of our computational approach.Except for the case of undirected circulant graphs of orders 27, the numerical results obtained here are new. In particular the number (up to isomorphism) of directed circulant graphs of orders 27, regardless of valency, is shown to be equal to 3,728,891 while 457 of these are self-complementary. Some curious and rather unexpected identities are established between intermediate valency-specified enumerators (both for undirected and directed circulant graphs) and their validity is conjectured for arbitrary cubed odd prime p 3 . We believe that this research can serve as the crucial step towards explicit uniform enumeration formulae for circulant graphs of orders p 3 for arbitrary prime p > 2.

show abstract

Counting circulant graphs of prime-power order by decomposing into orbit enumeration problems

Cited by 14 publications

References 0 publications

Enumeration Techniques on Cyclic Schur Rings

Enumeration Techniques on Cyclic Schur Rings

Counting Schur rings over cyclic groups

Constructive and analytic enumeration of circulant graphs with $p^3$ vertices; $p=3,5$

Contact Info

Product

Resources

About