Bibliographic record
Abstract
This thesis reviews some of the major results in the study of expander graphs.In particular this thesis will provide proofs of the Cheeger inequality and of the Alon-Boppana lower bound, the later leading naturally to study of Ramanujan graphs.The relationship between expander graphs and covering spaces will be explored, leading to a generalized notion of Ramanujan graphs.Connections between the matching polynomial and characteristic polynomial of a graph will be demonstrated and these connections will be applied in our presentation of a recent result of Marcus, Spielman and Srivastava which shows there exists Ramanujan families of all degrees.Several well known constructions of expander graphs will be described throughout this exposition, including a variant of the first explicitly constructed family of expander graphs introduced by Margulis in 1975.Some time will also be spent describing the first construction of families of Ramanujan graphs given by Lubotzky, Phillips, Sarnak in 1988.Throughout this review the reader will be exposed to some beautiful connections between expander graphs and other areas of mathematics including number theory, group theory, graph theory and basic linear algebra.This exposition hopes to serve as an accessible and interesting introduction to the known theory of expander graphs.iii ABRG Cette thse examine certains rsultats principaux dans l'tude des graphes expanseurs.En particulier, on prsente les preuves de l'ingalit de Cheeger et du thorme d'Alon-Boppana, ce dernier nous amne naturellement l'tude des graphes de Ramanujan.Nous allons expliquer les relations entre les graphes expanseurs et leurs revtements et dfinir une version gnralise de la notion de graphes de Ramanujan.On montrera en dtail comment le polynme caractristique et le polynme de couplage d'un graphe sont relis.On profite de ces liens pour prsenter un rsultat rcent de Marcus, Spielman et Srivastava qui affirme l'existence de familles de graphes de Ramanujan de tous degrs.Plusieurs constructions bien connues de familles de graphes expanseurs sont explicites dans cette thse, y compris une variante de la premire construction introduite par Margulis en 1975.Nous dcrivons aussi la premire construction de familles de graphes de Ramanujan introduite en 1988 par Lubotzky, Philips et Sarnak.Tout au long de ce travail, nous montrons comment l'tude des graphes expanseurs combine magnifiquement de nombreux domaines mathmatiques, y compris la combinatoire, la thorie des reprsentations, la thorie des groupes et la thorie des nombres.Cette thse se veut tre une introduction accessible l'tude des graphes expanseurs.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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 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".