2018/01/03 by Aida Abiad, Abiad, Aida, Steve Butler +3
Engineering · Mathematics · #05B20 #05C50 #05E30 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1801.01149
openalex publication_date 2018/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the behaviour 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 4m. 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 behaviour 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 4m for which the related strongly regular graphs have an unbounded number of different 2-ranks. The paper extends results from the article 'Switched symplectic graphs and their 2-ranks' by the first and the last author.