Open Problems on Graph Eigenvalues Studied with AutoGraphiX
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Bibliographic record
Abstract
Since the late forties of the last century, methods of operations research have been extensively used to solve problems in graph theory, and graph theory has been extensively used to model operations research problems and to solve optimization problems on graphs, e.g., shortest paths and network flow problems. More recently, methods of operations research and artificial intelligence have been used to advance graph theory per se, i.e., to find conjectures on graph theory invariants, to refute such conjectures and in some cases to find automated proofs or ideas of proofs. Among other systems, the AutoGraphiX system was developed since 1997 at GERAD (Montreal) by the present authors. Extensive experiments have been conducted which led to 1,700 conjectures, about 800 of which turned out to be easy and could be proved by the system, and about 600 further ones were proved by hand by us or graph theorists from various countries. Moreover, these results led to many generalizations and further papers. In this paper, we study four theoretical problems related to the eigenvalues of (the adjacency matrix of) a connected graph and to which AutoGraphiX was applied. Three of the problems are related to the maximum value of the irregularity, the maximum spectral spread and the upper bound of Nordhaus–Gaddum type on the index, over the class of connected graphs on $$n$$ vertices. The fourth problem concerns the maximization of the energy (the sum of the absolute values of the eigenvalues) of a connected graph with fixed numbers of vertices and of cycles. We present a brief survey of the papers on or in connection with these problems, and give some new partial results.
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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 | 0.001 | 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 | 0.001 | 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 it