Bibliographic record
Abstract
Abstract One way to study certain classes of polynomials is by considering examples that are attached to combinatorial objects. Any graph $G$ has an associated reciprocal polynomial $R_{G}$ , and with two particular classes of reciprocal polynomials in mind one can ask the questions: (a) when is $R_{G}$ a product of cyclotomic polynomials (giving thecyclotomic graphs)? (b) when does $R_{G}$ have the minimal polynomial of a Salem number as its only non-cyclotomic factor (the non-trivialSalem graphs)? Cyclotomic graphs were classified by Smith (Combinatorial structures and their applications, Proceedings of Calgary International Conference, Calgary, AB, 1969 (eds R. Guy, H. Hanani, H. Saver and J. Schönheim; Gordon and Breach, New York, 1970) 403–406); the maximal connected ones are known as Smith graphs. Salem graphs are ‘spectrally close’ to being cyclotomic, in that nearly all their eigenvalues are in the critical interval $[-2,2]$ . On the other hand, Salem graphs do not need to be ‘combinatorially close’ to being cyclotomic: the largest cyclotomic induced subgraph might be comparatively tiny. We define an $m$ -Salem graph to be a connected Salem graph $G$ for which $m$ is minimal such that there exists an induced cyclotomic subgraph of $G$ that has $m$ fewer vertices than $G$ . The $1$ -Salem subgraphs are both spectrally close and combinatorially close to being cyclotomic. Moreover, every Salem graph contains a $1$ -Salem graph as an induced subgraph, so these $1$ -Salem graphs provide some necessary substructure of all Salem graphs. The main result of this paper is a complete combinatorial description of all $1$ -Salem graphs: in the non-bipartite case there are $25$ infinite families and $383$ sporadic examples.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".