Balancing permuted copies of multigraphs and integer matrices

Coen del Valle, Peter J. Dukes*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Given a square matrix A over the integers, we consider the Z-module MA 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 MA. 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 λKn has an edge-decomposition into a given (multi)graph.

Original languageEnglish
Article number105756
Number of pages31
JournalJournal of Combinatorial Theory. Series A
Early online date12 Apr 2023
Publication statusPublished - 1 Aug 2023


  • Combinatorial matrix theory
  • Graph decomposition
  • Permutation


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