From Schur Rings to Constructive and Analytical Enumeration of Circulant Graphs with Prime-Cubed Number of Vertices

Victoria Gatt, Mikhail Klin, Josef Lauri, Valery Liskovets

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

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 and generating functions. The first method is based on the classification of circulant graphs in terms of S-rings and results in exhaustive listing (with the aid of COCO and GAP) of all corresponding S-rings of the indicated orders. The latter method is based on a general theoretical 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 graphs of such orders. In particular, five intermediate enumeration subproblems arise, which are refined further into eleven subproblems of this type. We give a brief survey of some background theory of the results which form the basis of our computational approach. Some curious and rather unexpected identities are established between intermediate valency-specified enumerators and their validity is conjectured for arbitrary cubed odd primes. We conclude with the enumeration of self-complementary circulant graphs of orders 27 and 125. We believe that this research can serve as the crucial step towards the explicit uniform enumeration formulae for circulant graphs of orders for arbitrary primes.

Original languageEnglish
Title of host publicationIsomorphisms, Symmetry and Computations in Algebraic Graph Theory, WAGT 2016
EditorsGareth A. Jones, Ilia Ponomarenko, Jozef Širán
PublisherSpringer
Pages37-65
Number of pages29
ISBN (Print)9783030328078
DOIs
StatePublished - 1 Jan 2020
EventInternational workshop on Isomorphisms, Symmetry and Computations in Algebraic Graph Theory, WAGT 2016 - Pilsen, Czech Republic
Duration: 3 Oct 20167 Oct 2016

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume305
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceInternational workshop on Isomorphisms, Symmetry and Computations in Algebraic Graph Theory, WAGT 2016
Country/TerritoryCzech Republic
CityPilsen
Period3/10/167/10/16

Keywords

  • Circulant graph
  • Combinatorial identity
  • Constructive enumeration
  • Cyclic group
  • Enumeration under group action
  • Graph isomorphism
  • Multiplier
  • Pólya’s method
  • S-ring
  • Self-complementary graphs

ASJC Scopus subject areas

  • Mathematics (all)

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