Switching checkerboards in (0,1)-matrices

David Ellison*, Bertrand Jouve, Lewi Stone

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In order to study M(R,C), the set of binary matrices with fixed row and column sums R and C, we consider submatrices of the form (1001) and (0110), called positive and negative checkerboard respectively. We define an oriented graph of matrices G(R,C) with vertex set M(R,C) and an arc from A to A indicates you can reach A by switching a negative checkerboard in A to positive. We show that G(R,C) is a directed acyclic graph and identify classes of matrices which constitute unique sinks and sources of G(R,C). Given A,A∈M(R,C), we give necessary conditions and sufficient conditions on M=A−A for the existence of a directed path from A to A. We then consider the special case of M(D), the set of adjacency matrices of graphs with fixed degree distribution D. We define G(D) accordingly by switching negative checkerboards in symmetric pairs. We show that Z2, an approximation of the spectral radius λ1 based on the second Zagreb index, is non-decreasing along arcs of G(D). Also, λ1 reaches its maximum in M(D) at a sink of G(D). We provide simulation results showing that applying successive positive switches to an Erdős-Rényi graph can significantly increase λ1.

Original languageEnglish
Pages (from-to)274-292
Number of pages19
JournalLinear Algebra and Its Applications
StatePublished - 1 Jan 2024
Externally publishedYes


  • Binary matrices
  • Bruhat order
  • Graph modifications
  • Graph theory
  • Spectral radius


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