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On the Clean Graph of a Ring

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Randhir Singh, S. C. Patekar

Combinatorics · Submitted 23 Jan 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper, we determine the Wiener index, Matching number of the clean graph of the ring Zn.

how it is stated · asserted 1.00 read the record →

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arXiv ID2301.09433 (version 1)
Pith IDYV7FETWA, Pith's permanent record of this paper and its review
CitationRandhir Singh, S. C. Patekar. "On the Clean Graph of a Ring." arXiv:2301.09433 (2023). https://pith.science/paper/YV7FETWA
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,580 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.29 · compares with prior results 0.03 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Let R be a ring (not necessarily commutative ring) with identity. The clean graph Cl(R) of a ring R is a graph with vertices in the form of ordered pair (e; u), where e is an idempotent of the ring R and u is a unit of the ring R. Two distinct vertices (e; u) and (f; v) are adjacent if and only if ef = fe = 0 or uv = vu = 1. In this paper, we determine the Wiener index, Matching number of the clean graph of the ring Zn.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let R be a ring (not necessarily commutative ring) with identity. background 1.00 0.04 0.01 asserted 0.85
s1 The clean graph Cl(R) of a ring R is a graph with vertices in the form of ordered pair (e; u), where e is an idempotent of the ring R and u is a unit of the ring R. background 0.94 0.10 0.02 asserted 1.00
s2 Two distinct vertices (e; u) and (f; v) are adjacent if and only if ef = fe = 0 or uv = vu = 1. background 0.92 0.09 0.03 asserted 1.00
s3 In this paper, we determine the Wiener index, Matching number of the clean graph of the ring Zn. contribution 0.94 0.96 0.76 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Let R be a ring (not necessarily commutative ring) with identity. The clean graph Cl(R) of a ring R is a graph with vertices in the form of ordered pair (e; u), where e is an idempotent of the ring R and u is a unit of the ring R. Two distinct vertices (e; u) and (f; v) are adjacent if and only if ef = fe = 0 or uv = vu = 1. In this paper, we determine the Wiener index, Matching number of the clean graph of the ring Zn.

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References
How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites it as background 1 paper

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