Bibliographic record
Abstract
AbstractIn this article, we characterize for which nite commutative ringR, the zero-divisor graph ( R), the line graph L(( R)), the com-plement graph ( R), and the line graph for the complement graphL(( R)) are Eulerian. Key Words: Zero-divisor graph, Eulerian graph, Local ring.2000 Mathematics Subject Classi cation: 13A99, 13M99, 05C15. 1 Introduction Let R be a nite commutative ring with 1. Z(R) is the set of zero-divisors ofR, and Z (R) = Z(R)nf0g. The zero-divisor graph of R, ( Z (R)), usuallywritten ( R), is the graph in which each element of Z (R) is a vertex, i.e.,V(( R)) = Z (R), and two distinct vertices x and y are adjacent if xy = 0.The complement graph ( R) is de ned on the same vertex set, but twodistinct vertices x and y are adjacent if xy 6= 0. For more properties ofthis graph, see [4] and [5]. The line graph of G, denoted by L(G), is agraph whose vertices are the edges of G, and two vertices of L(G) areadjacent whenever the corresponding edges of G are, see [7]. It is clear thatif G is connected, then so is L(G), while if L(G) is connected and G hasno isolated vertices, then G is also connected. If v;w are vertices in Gand vw is an edge joining them in G, then the degree of vw in L(G) isdeg(vw) = deg(v)+deg(w) 2. For distinct vertices x and y of a graph G,1
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 | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.005 | 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".