@inproceedings{c1433986cf2f43ef92672b49def084be,

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

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–P{\'o}lya 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.",

keywords = "Circulant graph, Combinatorial identity, Constructive enumeration, Cyclic group, Enumeration under group action, Graph isomorphism, Multiplier, P{\'o}lya{\textquoteright}s method, S-ring, Self-complementary graphs",

author = "Victoria Gatt and Mikhail Klin and Josef Lauri and Valery Liskovets",

note = "Funding Information: Acknowledgements The authors would like to thank Matan Ziv-Av for several helpful suggestions in the writing of GAP programs and for verifying some of our numerical results using diverse methods. The authors gratefully acknowledge support from the Scientific Grant Agency of the Slovak Republic under the number VEGA-1/0988/16. Publisher Copyright: {\textcopyright} 2020, Springer Nature Switzerland AG.; International workshop on Isomorphisms, Symmetry and Computations in Algebraic Graph Theory, WAGT 2016 ; Conference date: 03-10-2016 Through 07-10-2016",

year = "2020",

month = jan,

day = "1",

doi = "10.1007/978-3-030-32808-5_2",

language = "English",

isbn = "9783030328078",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer",

pages = "37--65",

editor = "Jones, {Gareth A.} and Ilia Ponomarenko and Jozef {\v S}ir{\'a}n",

booktitle = "Isomorphisms, Symmetry and Computations in Algebraic Graph Theory, WAGT 2016",

}