Switched symplectic graphs and their 2-ranks

Aida Abiad, W. H. Haemers

Research output: Contribution to journalArticleScientificpeer-review

Abstract

We apply Godsil–McKay switching to the symplectic graphs over F2 with at least 63 vertices and prove that the 2-rank of (the adjacency matrix of) the graph increases after switching. This shows that the switched graph is a new strongly regular graph with parameters (22ν−1,22ν−1,22ν−2,22ν−2) and 2-rank 2ν+2 when ν≥3 . For the symplectic graph on 63 vertices we investigate repeated switching by computer and find many new strongly regular graphs with the above parameters for ν=3 with various 2-ranks. Using these results and a recursive construction method for the symplectic graph from Hadamard matrices, we obtain several graphs with the above parameters, but different 2-ranks for every ν≥3 .
Original languageEnglish
Pages (from-to)35-41
JournalDesigns, Codes and Cryptography
Volume81
Issue number1
DOIs
Publication statusPublished - 2016

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Graph in graph theory
Hadamard matrices
Strongly Regular Graph
Hadamard Matrix
Adjacency Matrix

Keywords

  • strongly regular graph
  • symplectic graphs
  • switching
  • 2-rank
  • Hadamard matrix

Cite this

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title = "Switched symplectic graphs and their 2-ranks",
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Switched symplectic graphs and their 2-ranks. / Abiad, Aida; Haemers, W. H.

In: Designs, Codes and Cryptography, Vol. 81, No. 1, 2016, p. 35-41.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Switched symplectic graphs and their 2-ranks

AU - Abiad, Aida

AU - Haemers, W. H.

PY - 2016

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N2 - We apply Godsil–McKay switching to the symplectic graphs over F2 with at least 63 vertices and prove that the 2-rank of (the adjacency matrix of) the graph increases after switching. This shows that the switched graph is a new strongly regular graph with parameters (22ν−1,22ν−1,22ν−2,22ν−2) and 2-rank 2ν+2 when ν≥3 . For the symplectic graph on 63 vertices we investigate repeated switching by computer and find many new strongly regular graphs with the above parameters for ν=3 with various 2-ranks. Using these results and a recursive construction method for the symplectic graph from Hadamard matrices, we obtain several graphs with the above parameters, but different 2-ranks for every ν≥3 .

AB - We apply Godsil–McKay switching to the symplectic graphs over F2 with at least 63 vertices and prove that the 2-rank of (the adjacency matrix of) the graph increases after switching. This shows that the switched graph is a new strongly regular graph with parameters (22ν−1,22ν−1,22ν−2,22ν−2) and 2-rank 2ν+2 when ν≥3 . For the symplectic graph on 63 vertices we investigate repeated switching by computer and find many new strongly regular graphs with the above parameters for ν=3 with various 2-ranks. Using these results and a recursive construction method for the symplectic graph from Hadamard matrices, we obtain several graphs with the above parameters, but different 2-ranks for every ν≥3 .

KW - strongly regular graph

KW - symplectic graphs

KW - switching

KW - 2-rank

KW - Hadamard matrix

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JO - Designs, Codes and Cryptography

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