## Abstract

Given a square matrix A over the integers, we consider the Z-module M_{A} generated by the set of all matrices that are permutation-similar to A. Motivated by analogous problems on signed graph decompositions and block designs, we are interested in the completely symmetric matrices aI+bJ belonging to M_{A}. We give a relatively fast method to compute a generator for such matrices, avoiding the need for a very large canonical form over Z. Several special cases are considered. In particular, the problem for symmetric matrices answers a question of Cameron and Cioabǎ on determining the eventual period for integers λ such that the λ-fold complete graph λK_{n} has an edge-decomposition into a given (multi)graph.

Original language | English |
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Article number | 105756 |

Number of pages | 31 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 198 |

Early online date | 12 Apr 2023 |

DOIs | |

Publication status | Published - 1 Aug 2023 |

## Keywords

- Combinatorial matrix theory
- Graph decomposition
- Permutation