## Abstract

The operation of switching a graph Gamma with respect to a subset X of the vertex set interchanges edges and non-edges between X and its complement, leaving the rest of the graph unchanged. This is an equivalence relation on the set of graphs on a given vertex set, so we can talk about the automorphism group of a switching class of graphs.

It might be thought that switching classes with many automorphisms would have the property that all their graphs also have many automorphisms. But the main theorem of this paper shows a different picture: with finitely many

exceptions, if a non-trivial switching class *S* has primitive automorphism group, then it contains a graph whose automorphism group is trivial. We also find all the exceptional switching classes; up to complementation, there are just six.

Original language | English |
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Pages (from-to) | 76-90 |

Number of pages | 15 |

Journal | Australasian Journal of Combinatorics |

Volume | 62 |

Issue number | 1 |

Early online date | 2 Apr 2015 |

Publication status | Published - Jun 2015 |

## Keywords

- Switching classes
- Automorphism groups
- Primitive groups