Balancing permuted copies of multigraphs and integer matrices

Coen del Valle, Peter J. Dukes

Research output: Working paperPreprint


Given a square matrix Α 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. 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 λKn has an edge-decomposition into a given (multi)graph.
Original languageEnglish
Number of pages23
Publication statusPublished - 3 Jan 2022


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