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Cospectral Graphs and Regular Orthogonal Matrices of Level 2


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  • Abiad Monge, A.
  • Haemers, W.H.

    (Tilburg University, Center for Economic Research)

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    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.

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    Bibliographic Info

    Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 2012-042.

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    Date of creation: 2012
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    Handle: RePEc:dgr:kubcen:2012042

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    Keywords: cospectral graphs; orthogonal matrices; switching;

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    1. Dam, E.R. van & Haemers, W.H. & Koolen, J.H., 2006. "Cospectral Graphs and the Generalized Adjacency Matrix," Discussion Paper 2006-31, Tilburg University, Center for Economic Research.
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