Co-eigenvector graphs

Except for the empty graph, we show that the orthogonal matrix X of the adjacency matrix A determines that adjacency matrix completely, but not always uniquely. The proof relies on interesting properties of the Hadamard product Ξ=X∘X. As a consequence of the theory, we show that irregular co-eigenvector graphs exist only if the number of nodes N≥6. Co-eigenvector graphs possess the same orthogonal eigenvector matrix X, but different eigenvalues of the adjacency matrix. Co-eigenvector graphs are the dual of co-spectral graphs, that share all eigenvalues of the adjacency matrix, but possess a different orthogonal eigenvector matrix. We deduce general properties of co-eigenvector graph and start to enumerate all co-eigenvector graphs on N=6 and N=7 nodes. Finally, we list many open problems.