### Abstract

Original language | English |
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Place of Publication | Tilburg |

Publisher | Operations research |

Number of pages | 16 |

Volume | 2012-042 |

Publication status | Published - 2012 |

### Publication series

Name | CentER Discussion Paper |
---|---|

Volume | 2012-042 |

### Fingerprint

### Keywords

- cospectral graphs
- orthogonal matrices
- switching

### Cite this

*Cospectral Graphs and Regular Orthogonal Matrices of Level 2*. (CentER Discussion Paper; Vol. 2012-042). Tilburg: Operations research.

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**Cospectral Graphs and Regular Orthogonal Matrices of Level 2.** / Abiad, A.; Haemers, W.H.

Research output: Working paper › Discussion paper › Other research output

TY - UNPB

T1 - Cospectral Graphs and Regular Orthogonal Matrices of Level 2

AU - Abiad, A.

AU - Haemers, W.H.

N1 - Subsequently published in the Electronic Journal of Combinatorics (2012) Pagination: 16

PY - 2012

Y1 - 2012

N2 - Abstract: For a graph Γ with adjacency matrix A, we consider a switching operation that takes Γ into a graph Γ' with adjacency matrix A', defined by A' = QtAQ, where Q is a regular orthogonal matrix of level 2 (that is, QtQ = I, Q1 = 1, 2Q is integral, and Q is not a permutation matrix). If such an operation exists, and Γ is nonisomorphic with Γ', then we say that Γ' is semi-isomorphic with Γ. Semiisomorphic graphs are R-cospectral, which means that they are cospectral and so are their complements. Wang and Xu [‘On the asymptotic behavior of graphs determined by their generalized spectra’, Discrete Math. 310 (2010)] expect that almost all pairs of R-cospectral graphs are semi-isomorphic. Regular orthogonal matrices of level 2 have been classified. By use of this classification we work out the requirements for this switching operation to work in case Q has one nontrivial indecomposable block of size 4, 6, 7 or 8. Size 4 corresponds to Godsil-McKay switching. The other cases provide new methods for constructions of R-cospectral graphs. For graphs with eight vertices all these constructions are carried out. As a result we find that, out of the 1166 graphs on eight vertices which are R-cospectral to another graph, only 44 are not semi-isomorphic to another graph.

AB - Abstract: For a graph Γ with adjacency matrix A, we consider a switching operation that takes Γ into a graph Γ' with adjacency matrix A', defined by A' = QtAQ, where Q is a regular orthogonal matrix of level 2 (that is, QtQ = I, Q1 = 1, 2Q is integral, and Q is not a permutation matrix). If such an operation exists, and Γ is nonisomorphic with Γ', then we say that Γ' is semi-isomorphic with Γ. Semiisomorphic graphs are R-cospectral, which means that they are cospectral and so are their complements. Wang and Xu [‘On the asymptotic behavior of graphs determined by their generalized spectra’, Discrete Math. 310 (2010)] expect that almost all pairs of R-cospectral graphs are semi-isomorphic. Regular orthogonal matrices of level 2 have been classified. By use of this classification we work out the requirements for this switching operation to work in case Q has one nontrivial indecomposable block of size 4, 6, 7 or 8. Size 4 corresponds to Godsil-McKay switching. The other cases provide new methods for constructions of R-cospectral graphs. For graphs with eight vertices all these constructions are carried out. As a result we find that, out of the 1166 graphs on eight vertices which are R-cospectral to another graph, only 44 are not semi-isomorphic to another graph.

KW - cospectral graphs

KW - orthogonal matrices

KW - switching

M3 - Discussion paper

VL - 2012-042

T3 - CentER Discussion Paper

BT - Cospectral Graphs and Regular Orthogonal Matrices of Level 2

PB - Operations research

CY - Tilburg

ER -