Cospectral graphs and regular orthogonal matrices of level 2

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 ́=Q ^⊤AQ, where Q is a regular orthogonal matrix of level 2(that is, Q ^⊤Q=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. Semi-isomorphic 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 nonisomorphic 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.