Switching checkerboards in (0,1)-matrices

Abstract

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.