Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-11T04:39:51.754092Z
Paper Citation Record · LEDGER
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability
As of 17 August 2026, this Paper Citation Record lists 23 of 23 outbound references and 0 inbound Pith citation observations for arXiv:2412.18759.
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 2026-08-17T06:30:58.91139+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
23 of 23 outbound references displayed
- verified exact0
- verified fuzzy17
- unresolved6
- parse uncertain0
- malformed identifier0
- metadata mismatch0
External citation measurements
No source-named external measurement is stored.
Outbound references
Observation ba2f1afb-9def-45f1-ad97-06a900fc9e44 · outbound
This paper cites A note on the eigenvalues of a graph.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability A note on the eigenvalues of a graph
Reference 1
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:52.179814Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 7d1bc4b9-e4ca-4024-b882-73ac558f01aa · outbound
This paper cites Linear System Theory and Design: International Fourth Edit ion.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Linear System Theory and Design: International Fourth Edit ion
Reference 2
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:52.161558Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 7b39edb8-2948-47c9-8d68-686f523626c2 · outbound
This paper cites Cvetkovi´ c, P.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Cvetkovi´ c, P
Reference 3
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:52.146722Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation ad6b11d6-99d8-40f5-8050-217d3a6d587d · outbound
This paper cites Controllable graphs with least eigenvalue at least −2.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Controllable graphs with least eigenvalue at least −2
Reference 4
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:52.132718Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation b2c6b1e1-34cc-40e0-8957-505cf11c04da · outbound
This paper cites Contr ollability of NEPSes of graphs.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Contr ollability of NEPSes of graphs
Reference 5
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:52.114328Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 42a2897d-b133-4b91-bb91-3705a0320422 · outbound
This paper cites an unresolved cited work.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Unresolved cited work
Reference 6
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T04:39:52.098254Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 5c9066d1-0bfc-4597-b19b-5dd7bbaa7b46 · outbound
This paper cites an unresolved cited work.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Unresolved cited work
Reference 7
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T04:39:52.086210Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 9f71fa66-6bf8-467e-b0f6-715ce292440e · outbound
This paper cites Haemers and G.R.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Haemers and G.R
Reference 8
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:52.073199Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 9d749a1f-13ea-458d-a362-c72e73e8c732 · outbound
This paper cites an unresolved cited work.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Unresolved cited work
Reference 9
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T04:39:52.060670Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 12255f2e-86a6-4615-9a6f-e2e4a3adc7fb · outbound
This paper cites Number of connected graphs on n vertices whose spectrum has n distinct eigenvalues.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Number of connected graphs on n vertices whose spectrum has n distinct eigenvalues
Reference 10
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:52.046981Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation fe6d2aee-33b1-4f9b-a62d-8300ac488cb1 · outbound
This paper cites On the hermit ian matrix with distinct eigenvalues and its applications.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability On the hermit ian matrix with distinct eigenvalues and its applications
Reference 11
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:52.021221Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 14925af1-7916-4ed3-b935-e0144defc452 · outbound
This paper cites Construc tion of graphs with distinct eigenvalues.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Construc tion of graphs with distinct eigenvalues
Reference 12
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:51.996155Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation c3d143c6-fecb-4919-a0bc-c58cd8fff034 · outbound
This paper cites On the con struction of Q-controllable graphs.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability On the con struction of Q-controllable graphs
Reference 13
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:51.967785Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 6ab1ff73-36b6-4546-8f19-7e6b0180fa15 · outbound
This paper cites Solving the Korteweg-de Vries eq uation by its bilinear form: Wronskian solutions.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Solving the Korteweg-de Vries eq uation by its bilinear form: Wronskian solutions
Reference 14
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:51.938096Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation c9b2cdbc-cdab-41d3-8708-24ae96460337 · outbound
This paper cites The signless Laplacian sp ectrum of rooted product of graphs.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability The signless Laplacian sp ectrum of rooted product of graphs
Reference 15
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:51.922804Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 8dc8560f-0d21-44cb-9232-11f425a8d888 · outbound
This paper cites The group of a graph whose adjacency matr ix has all distinct eigenvalues.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability The group of a graph whose adjacency matr ix has all distinct eigenvalues
Reference 16
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:51.906971Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 4b783e55-f8fd-49c1-ad8b-79eb08975d27 · outbound
This paper cites Rajkumar and R.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Rajkumar and R
Reference 17
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:51.883024Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation c198c8fc-e084-4bb8-9c44-eb65736b8e8a · outbound
This paper cites an unresolved cited work.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Unresolved cited work
Reference 18
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T04:39:51.863521Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation d3b827f7-ace3-41f0-9c91-17c9608c6ebe · outbound
This paper cites Polynomials, roots, and interlacing.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Polynomials, roots, and interlacing
Reference 19
- Resolution
- unresolved
- no resolver link, observed 2026-08-11T04:39:51.697542Z
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 7e938d8c-2c23-43af-b914-dc04e1b4b3b4 · outbound
This paper cites Construction of gra phs with distinct Aα -eigenvalues.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Construction of gra phs with distinct Aα -eigenvalues
Reference 20
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:51.851357Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation b7c57c73-1340-43e2-aabd-90a5d9a37e96 · outbound
This paper cites an unresolved cited work.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability Unresolved cited work
Reference 21
- Resolution
- unresolved
- raw_fallback, observed 2026-08-11T04:39:51.837789Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 8fb7a2f9-76ef-4b55-b2e3-83e6b6436b2c · outbound
This paper cites The controllability of graphs with diameter n-2.
Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability The controllability of graphs with diameter n-2
Reference 22
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:51.817223Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Observation 3f1398ce-c305-404a-97f1-b042cf651b59 · outbound
This paper cites Weisstein.
Reference 23
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-11T04:39:51.801533Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.
Pith citing papers
No inbound Pith citation observations are available.