Balancing permuted copies of multigraphs and integer matrices

TL;DR

This paper studies the structure of integer matrices generated by permutation similarities of a given matrix, focusing on symmetric matrices of the form aI + bJ, with applications to graph decompositions.

Contribution

It introduces a fast method to compute generators for symmetric matrices in the module, addressing a problem related to graph decompositions and integer matrices.

Findings

Developed a quick algorithm for symmetric matrix generation

Solved a problem on graph edge-decompositions into multigraphs

Provided detailed analysis of special cases in the module

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 Cioab\v{a} on determining the eventual period for integers $\lambda$ such that the $\lambda$-fold complete graph $\lambda K_n$ has an edge-decomposition into a given (multi)graph.