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arXiv · 2201.00897

Balancing permuted copies of multigraphs and integer matrices

Coen del Valle·Peter J. Dukes
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Abstract

Given a square matrix $A$ over the integers, we consider the $\mathbb{Z}$-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 $a I + b J$ 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 $\mathbb{Z}$. We consider several special cases in detail. 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.

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Coen del Valle, Peter J. Dukes. 2026-06-11. Balancing permuted copies of multigraphs and integer matrices. https://arxiv.org/abs/2201.00897

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