Realizing the Asymmetric Index of a Graph
Bibliographic record
Abstract
A graph G is asymmetric if its automorphism group is trivial. Asymmetric graphs were introduced by Erd\H{o}s and R\'{e}nyi Erdos [1]. They suggested the problem of starting with an asymmetric graph and removing some number, r , of edges and/or adding some number, s , of edges so that the resulting graph is non-asymmetric. Erd\H{o}s and R\'{e}nyi defined the degree of asymmetry of a graph to be the minimum value of r + s . In this paper, we consider another property that measures how close a given non-asymmetric graph is to being asymmetric. Brewer et al. defined the asymmetric index of a graph G , denoted a i ( G ) , as the minimum of r + s so that the resulting graph G is asymmetric [2]. It is noted that a i ( G ) is only defined for graphs with at least six vertices. We investigate the asymmetric index of both connected and disconnected graphs including paths, cycles, and grids, with the addition of up to two isolated vertices. Furthermore, for a graph in these families G , we determine the number of labeled asymmetric graphs that can be obtained by adding or removing a i ( G ) edges. This leads to the related question: Given a graph G where a i ( G ) = 1 , what is the probability that for a randomly chosen edge e , that G – e will be asymmetric? A graph is called minimally non-asymmetric if this probability is 1 . We give a construction of infinite families of minimally non-asymmetric graphs.
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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 | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".