### 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.

Original language | English |
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Pages (from-to) | 171-182 |

Number of pages | 12 |

Journal | Annals of Combinatorics |

Volume | 23 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2019 |

## Cite this

*Annals of Combinatorics*,

*23*(1), 171-182. https://doi.org/10.1007/s00026-019-00423-0