Paper Citation Record · LEDGER

Absence of flat bands for discrete periodic graph operators with generic potentials

As of 15 August 2026, this Paper Citation Record lists 36 of 36 outbound references and 0 inbound Pith citation observations for arXiv:2509.01927.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2509.01927 v1

Coverage vector

measured 36 of 36 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-05T12:17:29.873924Z

measured 36 of 36 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-14T06:32:32.682623+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

36 of 36 outbound references displayed

verified exact4
verified fuzzy14
unresolved18
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 8e81b557-1718-4539-8285-ec36aebcc49e · outbound

This paper cites Nauk 38 (1983), no.

Absence of flat bands for discrete periodic graph operators with generic potentials Nauk 38 (1983), no

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:31.308711Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 81387ffb-d7e0-471f-a1c8-367703c9fd1b · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:31.278898Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 9418f0bf-2c19-4b78-b03e-6eb3ee805ae5 · outbound

This paper cites Sh., Suslina T.

Absence of flat bands for discrete periodic graph operators with generic potentials Sh., Suslina T

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:31.258839Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 62fda2b7-3ffe-4479-98a4-28b4866ac9fd · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 4

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:31.220688Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 36f05517-1e26-4596-865c-6c1b5b62bf6a · outbound

This paper cites Rare Flat Bands for Periodic Graph Operators.

Absence of flat bands for discrete periodic graph operators with generic potentials Rare Flat Bands for Periodic Graph Operators

Reference 5

Resolution: verified exact; local_arxiv, observed 2026-08-05T12:17:30.584869Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation fe72587a-52a1-46c6-8272-7404fc75421d · outbound

This paper cites L., Irreducibility of the Dispersion Polynomial for Periodic Graphs , SIAM Journal on Applied Algebra and Geometry 9, (2025), no.

Absence of flat bands for discrete periodic graph operators with generic potentials L., Irreducibility of the Dispersion Polynomial for Periodic Graphs , SIAM Journal on Applied Algebra and Geometry 9, (2025), no

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:31.196181Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 261df3c0-15a3-4d40-9af0-37e473aef189 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 7

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:31.172201Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 65d0dbdb-3800-4b85-9459-cebbaf339670 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 8

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:31.144852Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 2b77f75d-6cc9-405c-b8e7-ac1e650103b6 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 9

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:31.110519Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 32c87c0e-c528-49a0-8c3f-88c76afbfff2 · outbound

This paper cites 2 (2006), 3826 – 3831.

Absence of flat bands for discrete periodic graph operators with generic potentials 2 (2006), 3826 – 3831

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:31.091396Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 4856923f-8523-4226-bb25-e1c14ea85cda · outbound

This paper cites P., Reducible Fermi surface for multi-layer quantum graphs including stacked graphene, Comm.

Absence of flat bands for discrete periodic graph operators with generic potentials P., Reducible Fermi surface for multi-layer quantum graphs including stacked graphene, Comm

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:31.070822Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 7d1daf4b-50f0-4532-9ec9-870dc09bdcfe · outbound

This paper cites 2, 570 – 585.

Absence of flat bands for discrete periodic graph operators with generic potentials 2, 570 – 585

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:31.047662Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 15fcbbd7-8e50-489e-8b99-dab78e0d028e · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 13

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:31.030289Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation b7e2b425-3e9f-47de-bda8-8a985b8f61a4 · outbound

This paper cites Henri Poincar´ e 25, (2024), no.

Absence of flat bands for discrete periodic graph operators with generic potentials Henri Poincar´ e 25, (2024), no

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:31.009193Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 5935e6b0-f8c1-43e4-b600-eefa755ea884 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 15

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.989274Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation b4f388a7-29ad-4f22-99f8-2c09d569a69f · outbound

This paper cites Periodic and limit-periodic discrete Schr\"odinger operators.

Absence of flat bands for discrete periodic graph operators with generic potentials Periodic and limit-periodic discrete Schr\"odinger operators

Reference 16

Resolution: verified exact; local_arxiv, observed 2026-08-05T12:17:30.508913Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 04277c8f-ad3e-4e26-a27f-fea60b959dfd · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 17

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.969092Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 7e56cd88-6387-44fd-804e-472bbfe2dc7a · outbound

This paper cites What is known and unknown , J.

Absence of flat bands for discrete periodic graph operators with generic potentials What is known and unknown , J

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:30.949690Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 21d16a30-8abd-495b-a53a-b18ef632c686 · outbound

This paper cites ABSENCE OF FLAT BANDS FOR DISCRETE PERIODIC GRAPH OPERATORS WITH GENERIC POTENTIALS 19.

Absence of flat bands for discrete periodic graph operators with generic potentials ABSENCE OF FLAT BANDS FOR DISCRETE PERIODIC GRAPH OPERATORS WITH GENERIC POTENTIALS 19

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:30.930824Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 453c1375-0073-40b9-aab6-8ffe7ff4d283 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 20

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.909575Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 8a5a31d0-870d-463d-ac5b-e9792113b1a3 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 21

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.889150Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 82b8d2ed-c5f5-411b-b868-d1ad0638b7a8 · outbound

This paper cites 9 – 10, 1809 – 1826.

Absence of flat bands for discrete periodic graph operators with generic potentials 9 – 10, 1809 – 1826

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:30.865854Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 22361d19-2f49-47ec-8db1-e5182e5bd2c7 · outbound

This paper cites 3, 673 – 686.

Absence of flat bands for discrete periodic graph operators with generic potentials 3, 673 – 686

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:30.844819Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation bc5f1b5e-9913-4d95-b31e-7819535d0aad · outbound

This paper cites P., Irreducibility of the Fermi surface for planar periodic graph operators , Lett.

Absence of flat bands for discrete periodic graph operators with generic potentials P., Irreducibility of the Fermi surface for planar periodic graph operators , Lett

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:30.826897Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation ea12006c-2201-4bca-9be6-3b04be991cb6 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 25

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.802281Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 7abdcf65-d293-470f-8cb5-63ff23b538ed · outbound

This paper cites Sharp decay rate for eigenfunctions of perturbed periodic Schr\"odinger operators.

Absence of flat bands for discrete periodic graph operators with generic potentials Sharp decay rate for eigenfunctions of perturbed periodic Schr\"odinger operators

Reference 27

Resolution: verified exact; local_arxiv, observed 2026-08-05T12:17:30.258780Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation e48d5a7d-5a15-4918-8bb3-0c3221ce1c51 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 28

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.781052Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation ecb43c4b-0b3b-4156-9c4f-d8b714db3ede · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 29

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.750406Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 6e7d5c99-4d30-46c9-87f9-c7e04f960aed · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 30

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.728968Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 13b9ef65-af56-42be-b4b2-d7bc9db16117 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 31

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.709234Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation a567e517-7071-4a08-b443-2aaa19308577 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 32

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.687762Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 7d2950f3-8da8-4cd8-9011-15f24a8671e3 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 33

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.666012Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 943fd28f-002c-4466-a178-138debd7b624 · outbound

This paper cites P., Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators , J.

Absence of flat bands for discrete periodic graph operators with generic potentials P., Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators , J

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:30.648438Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 48b16e1e-ae0f-4445-b16c-f326ba3dcdc6 · outbound

This paper cites P., Eigenfunctions of unbounded support for embedded eigenvalues of locally perturbed periodic graph operators, Comm.

Absence of flat bands for discrete periodic graph operators with generic potentials P., Eigenfunctions of unbounded support for embedded eigenvalues of locally perturbed periodic graph operators, Comm

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T12:17:30.625837Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 0388d72c-0066-42a2-932d-d3a76038dc16 · outbound

This paper cites Algebraic Aspects of Periodic Graph Operators.

Absence of flat bands for discrete periodic graph operators with generic potentials Algebraic Aspects of Periodic Graph Operators

Reference 36

Resolution: verified exact; local_arxiv, observed 2026-08-05T12:17:30.009293Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation f20542dd-7bd0-49b1-bb7d-5c42b8531e44 · outbound

This paper cites an unresolved cited work.

Absence of flat bands for discrete periodic graph operators with generic potentials Unresolved cited work

Reference 37

Resolution: unresolved; raw_fallback, observed 2026-08-05T12:17:30.605766Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Pith citing papers

No inbound Pith citation observations are available.