Graph switching, 2-ranks, and graphical Hadamard matrices

We study the behavior of the 2-rank of the adjacency matrix of a graph under Seidel and Godsil-McKay switching, and apply the result to graphs coming from graphical Hadamard matrices of order 4(m). Starting with graphs from known Hadamard matrices of order 64, we find (by computer) many Godsil-McKay switching sets that increase the 2-rank. Thus we find strongly regular graphs with parameters (63, 32, 16, 16), (64, 36, 20, 20), and (64, 28, 12, 12) for almost all feasible 2-ranks. In addition we work out the behavior of the 2-rank for a graph product related to the Kronecker product for Hadamard matrices, which enables us to find many graphical Hadamard matrices of order 4(m) for which the number of related strongly regular graphs with different 2-ranks is unbounded as a function of m. The paper extends results from the article 'Switched symplectic graphs and their 2-ranks' by the first and the last author. (C) 2018 Elsevier B.V. All rights reserved.