### Abstract

Original language | English |
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Journal | Discrete Mathematics |

DOIs | |

Publication status | E-pub ahead of print - Dec 2018 |

### Fingerprint

### Keywords

- strongly regular graph
- Seidel switching
- Godsil–McKay switching
- 2-rank
- Hadamard matrix

### Cite this

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**Graph switching, 2-ranks, and graphical Hadamard matrices.** / Abiad, Aida; Butler, Steve; Haemers, Willem H.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Graph switching, 2-ranks, and graphical Hadamard matrices

AU - Abiad, Aida

AU - Butler, Steve

AU - Haemers, Willem H.

PY - 2018/12

Y1 - 2018/12

N2 - 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 $4^m$. 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 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 $4^m$ 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.

AB - 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 $4^m$. 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 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 $4^m$ 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.

KW - strongly regular graph

KW - Seidel switching

KW - Godsil–McKay switching

KW - 2-rank

KW - Hadamard matrix

U2 - 10.1016/j.disc.2018.11.022

DO - 10.1016/j.disc.2018.11.022

M3 - Article

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

ER -