Abstract
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. Coeigenvector 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.
Original language | English |
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Pages (from-to) | 34-59 |
Number of pages | 26 |
Journal | Linear Algebra and Its Applications |
Volume | 689 |
DOIs | |
Publication status | Published - 2024 |
Keywords
- graph spectra
- eigenvectors
- eigenvalues
- co-eigenvector graphs