Balancing permuted copies of multigraphs and integer matrices

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 $aI+bJ$ 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 $\mathbb{Z}$. 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 $\hat{I}»$ such that the $\hat{I}»$-fold complete graph $\hat{I}»K_n$ has an edge-decomposition into a given (multi)graph.