Graph switching, 2-ranks, and graphical Hadamard matrices

Aida Abiad Monge, S. Butler, Willem Haemers*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


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.
Original languageEnglish
Pages (from-to)2850-2855
Number of pages6
JournalDiscrete Mathematics
Issue number10
Publication statusPublished - Oct 2019
EventAlgebraic and Extremal Graph Theory Conference - University of Delaware, Newark, United States
Duration: 7 Aug 201710 Aug 2017


  • Strongly regular graph
  • Seidel switching
  • Godsil-McKay switching
  • 2-rank
  • Hadamard matrix

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