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.

pith.paper-citation-record.v1
2608.07880 v1

Coverage vector

measured 11 of 11 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-12T00:56:27.065694Z

measured 11 of 11 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-21T06:32:19.484+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+00:00.

Pith citing papers

No inbound Pith citation observations are available.