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Preprint · Version 1Graph theory and discrete mathematicsPreprint; not peer reviewed

A Counterexample to a Line-Graph Inertia Conjecture

Zenodo deposit
Version
1.0

Abstract

The work presents a connected simple graph whose line graph has adjacency inertia (9, 0, 7), giving an exact counterexample to Conjecture 4.12. The public record includes an adjacency matrix, exact-arithmetic certificates, and independent scripts reproducing the inertia computation and incidence congruence.

Key formulae

Inertia⁡(A(L(G)))=(9,0,7)\operatorname{Inertia}(A(L(G)))=(9,0,7)

Document status

Version 1.0, preserved as the historical finite-counterexample record. It has not undergone peer review and is not presented as a journal publication.

Main contributions

  • An explicit, finite, connected graph that refutes the conjecture: in its line graph 9 eigenvalues are positive, none is zero and 7 are negative.
  • The certificates of the computation in exact arithmetic, with no approximation, and the graph's matrix published in full.
  • Two independent programs: one computes directly, the other checks by a different route.

Stated limitations

  • This version establishes a single counterexample. The infinite family and the means to carry the result belong to Version 2.
  • This is a preprint: no journal has reviewed it.

Versions and provenance

Among the public records currently identified, the Version 1 Zenodo deposit dated 22 July 2026 predates later identified related public preprints. No claim is made about undiscovered, private, unpublished, or otherwise unidentified antecedents.

Related work

Verifiable provenance

The public repository and recorded Git objects are also preserved by Software Heritage.

Public materials

Technical details
Authoritative archive
Zenodo
Date of this file
This file has existed since at least 28 July 2026. The proof sits in a public register we do not control, and anyone can check it.
SHA-256
ae84a56899307dd7247ee113b437f88d41f965464719d1c6d662d115be3af1d4
SHA-256 · Zenodo reproducibility package
2c77be7ed293edfeec7f57654e1442a72fc1a172f33d059a890a6fa54d7c39e0
Timestamp receipt (.ots)
Timestamp receipt (.ots)
Bitcoin blocks
959991, 959992, 960019, 960020
Verification record
AVR-AT-SP-2026-001-v1.0
Where to find this work elsewhere

Citation

BibTeX RIS
BibTeX 
@misc{Paone2026ACounterexampleLine,
  author = {Paone, Andrea},
  title = {A Counterexample to a Line-Graph Inertia Conjecture},
  month = jul,
  year = {2026},
  date = {2026-07-22},
  version = {1.0},
  doi = {10.5281/zenodo.21499790},
  url = {https://doi.org/10.5281/zenodo.21499790},
  note = {Preprint; not peer reviewed}
}
RIS 
TY  - UNPB
AU  - Paone, Andrea
TI  - A Counterexample to a Line-Graph Inertia Conjecture
PY  - 2026
DA  - 2026-07-22
DB  - Zenodo
ET  - 1.0
DO  - 10.5281/zenodo.21499790
UR  - https://doi.org/10.5281/zenodo.21499790
N1  - Preprint; not peer reviewed
KW  - line graph
KW  - graph inertia
KW  - inertia indices
KW  - spectral graph theory
KW  - counterexample
KW  - Conjecture 4.12
KW  - adjacency matrix
KW  - characteristic polynomial
KW  - exact arithmetic
KW  - incidence congruence
KW  - reproducibility certificate
ER  -

Keywords