Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-12T00:56:27.065694Z
Paper Citation Record · LEDGER
Sharp vertex connectivity of the Markoff graphs modulo $p$
As of 22 August 2026, this Paper Citation Record lists 11 of 11 outbound references and 0 inbound Pith citation observations for arXiv:2608.07880.
A citation records a reference. It does not transfer a finding from one paper to another.
Coverage vector
One-hop event checks from named stored sources.
Source: scholarly_work_events, retraction_status_cache, observed 2026-08-22T06:32:14.747728+00:00
Pith citing papers itemized under the disclosed page cap.
Source: paper_references, paper_reference_links
A source-named dated measurement, never combined with another source.
Source: cited_works
Reference resolution
11 of 11 outbound references displayed
- verified exact0
- verified fuzzy8
- unresolved3
- parse uncertain0
- malformed identifier0
- metadata mismatch0
External citation measurements
No source-named external measurement is stored.
Outbound references
Observation 00f04130-5353-4b78-975c-1954ecccedc3 · outbound
This paper cites Baragar.The Markoff Equation and Equations of Hurwitz.
Sharp vertex connectivity of the Markoff graphs modulo $p$ Baragar.The Markoff Equation and Equations of Hurwitz
Reference 1
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-12T00:56:27.281759Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 132bb7f6-96b6-4566-9bc0-50e0a412d004 · outbound
This paper cites Markoff triples and strong approximation.Comptes Rendus Math´ ematique, 354(2):131–135, 2016.
Sharp vertex connectivity of the Markoff graphs modulo $p$ Markoff triples and strong approximation.Comptes Rendus Math´ ematique, 354(2):131–135, 2016
Reference 2
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-12T00:56:27.266312Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 040202c0-1798-4aa3-80f9-1016ea5a7e7c · outbound
This paper cites Strong approximation and Diophantine properties of Markoff triples.Journal of the American Mathematical Society, 39(1):177–204, 2026.
Sharp vertex connectivity of the Markoff graphs modulo $p$ Strong approximation and Diophantine properties of Markoff triples.Journal of the American Mathematical Society, 39(1):177–204, 2026
Reference 3
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-12T00:56:27.250149Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 7a34603f-51a9-430f-a552-48765311a763 · outbound
This paper cites An almost linear time algorithm testing whether the Markoff graph modulopis connected.Research in Number Theory, 11(1), 2025.
Sharp vertex connectivity of the Markoff graphs modulo $p$ An almost linear time algorithm testing whether the Markoff graph modulopis connected.Research in Number Theory, 11(1), 2025
Reference 4
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-12T00:56:27.234297Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 1dbb8848-ff7b-45f9-8242-5996bfe01a85 · outbound
This paper cites The number of points on certain cubic surfaces over a finite field.Bollettino dell’Unione Matematica Italiana, 12(1):19–21, 1957.
Sharp vertex connectivity of the Markoff graphs modulo $p$ The number of points on certain cubic surfaces over a finite field.Bollettino dell’Unione Matematica Italiana, 12(1):19–21, 1957
Reference 5
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-12T00:56:27.217216Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 1166dd7e-f25a-4b13-841c-3fb3f59d6cde · outbound
This paper cites The cycle structure of a Markoff automorphism over finite fields.Journal of Number Theory, 211:1–27, 2020.
Sharp vertex connectivity of the Markoff graphs modulo $p$ The cycle structure of a Markoff automorphism over finite fields.Journal of Number Theory, 211:1–27, 2020
Reference 6
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-12T00:56:27.199341Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 78cb1852-3289-4896-8918-ced48b5be2fe · outbound
This paper cites an unresolved cited work.
Sharp vertex connectivity of the Markoff graphs modulo $p$ Unresolved cited work
Reference 7
- Resolution
- unresolved
- raw_fallback, observed 2026-08-12T00:56:27.180111Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation e844ce12-86ed-4da3-abb4-727f610b3fdf · outbound
This paper cites Non-planarity of Markoff graphs mod p.Commentarii Mathematici Helvetici, 99(1):111-148, 2024.
Sharp vertex connectivity of the Markoff graphs modulo $p$ Non-planarity of Markoff graphs mod p.Commentarii Mathematici Helvetici, 99(1):111-148, 2024
Reference 8
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-12T00:56:27.162799Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 6b2d0347-3feb-4389-be66-394830e7ea7d · outbound
This paper cites Martin, and Nico Tripeny.
Sharp vertex connectivity of the Markoff graphs modulo $p$ Martin, and Nico Tripeny
Reference 9
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-12T00:56:27.144680Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 92b260e0-c1a4-4198-abfc-ca6b09f51de8 · outbound
This paper cites an unresolved cited work.
Sharp vertex connectivity of the Markoff graphs modulo $p$ Unresolved cited work
Reference 10
- Resolution
- unresolved
- raw_fallback, observed 2026-08-12T00:56:27.127402Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 6c6b2b89-0712-4b9b-b665-f3c110567cda · outbound
This paper cites an unresolved cited work.
Sharp vertex connectivity of the Markoff graphs modulo $p$ Unresolved cited work
Reference 11
- Resolution
- unresolved
- raw_fallback, observed 2026-08-12T00:56:27.109132Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Pith citing papers
No inbound Pith citation observations are available.