Cospectral graphs and regular orthogonal matrices of level 2 (Q456333)

Summary: For a graph \(\Gamma\) with adjacency matrix \(A\), we consider a switching operation that takes \(\Gamma\) into a graph \(\Gamma'\) with adjacency matrix \(A'\), defined by \(A'=Q^\top A Q\), where \(Q\) is a regular orthogonal matrix of level 2 (that is, \(Q^\top Q=I\), \(Q1 = 1\), \(2Q\) is integral, and \(Q\) is not a permutation matrix). If such an operation exists, and \(\Gamma\) is nonisomorphic with \(\Gamma'\), then we say that \(\Gamma'\) is semi-isomorphic with \(\Gamma\). Semi-isomorphic graphs are \(\mathbb {R}\)-cospectral, which means that they are cospectral and so are their complements. \textit{W. Wang} and \textit{C.-H. Xu} [Linear Algebra Appl. 418, No. 1, 62--74 (2006; Zbl 1105.05050)] expect that almost all pairs of nonisomorphic \(\mathbb {R}\)-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 \(\mathbb {R}\)-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 \(\mathbb {R}\)-cospectral to another graph, only 44 are not semi-isomorphic to another graph.