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Generalised Paley Graphs with a Product Structure

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Abstract

A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the Cartesian-prime factors of the Cartesian decomposition, and were shown, by Sabidussi and Vizing independently, to be uniquely determined up to isomorphism. We characterise by their parameters those generalised Paley graphs which are Cartesian decomposable, and we prove that for such graphs, the Cartesian-prime factors are themselves smaller generalised Paley graphs. This generalises a result of Lim and the second author which deals with the case where all the Cartesian-prime factors are complete graphs. These results contribute to the determination, by parameters, of generalised Paley graphs with automorphism groups larger than the one-dimensional affine subgroups used to define them. They also shed light on the structure of primitive cyclotomic association schemes.

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Notes

  1. This conjecture is a modification of a conjecture of Brouwer [3, Conjecture, Page 68] for such schemes, which had been disproved by Brouwer himself [2].

  2. See https://www.win.tue.nl/~aeb/graphs/VO.html.

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Acknowledgements

We are grateful to Gareth Jones for helpful discussions about the origin of the name Paley graphs. We are also grateful for the nudge he gave us to write up our work for publication. The beginnings of this investigation go back to an undergraduate research project of G. Pearce. In addition the authors are grateful for the thoughtful and helpful comments of anonymous referees which improved the exposition, and also led to a strengthening of Lemma 3.3.

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Authors and Affiliations

  1. Perth Modern School, 90 Roberts Road, Subiaco, WA, 6008, Australia

    Geoffrey Pearce

  2. Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Cheryl E. Praeger

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  1. Geoffrey Pearce
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  2. Cheryl E. Praeger
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Correspondence to Geoffrey Pearce.

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Pearce, G., Praeger, C.E. Generalised Paley Graphs with a Product Structure. Ann. Comb. 23, 171–182 (2019). https://doi.org/10.1007/s00026-019-00423-0

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