Eigenvalue and Eigenvector Information of Graphs and Their Validity in Detection of Graph Isomorphism
Bibliographic record
Abstract
Detection of graph isomorphism (GI) has been widely used in many fields in science and engineering. Currently, a potential application of GI detection could be in molecular structure design for microelectromechanical systems and nano-systems. In this paper, we discuss the relationship between graphs and their eigenvalues as well as unique eigenvectors. We prove that the graphs having all distinct eigenvalues are isomorphic if and only if they have the same graph spectrum and the equivalent eigenvectors. The graphs having coincident eigenvalues might be isomorphic if they have the same graph spectrum and the equivalent unique eigenvectors. Further, a convergent recursive procedure is given to subdivide a group-to-group mapping once appeared in the graphs having coincident eigenvalues to seek potential one-to-one mappings so as to determining if the graphs are isomorphic.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".