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Preprint · Version 2.0-rev2Graph theory and discrete mathematicsPreprint; not peer reviewed

Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

Zenodo deposit
Version
2.0-rev2

Abstract

Akbari, Elphick, Kumar, Pragada and Tang conjectured that every connected graph satisfies a one-unit upper bound between the positive and negative adjacency inertia indices of its line graph. The manuscript presents a connected simple counterexample with line-graph inertia (9, 0, 7) and proves that the conjecture fails without bound even for connected simple planar subcubic cactus graphs. It introduces a rooted-module attachment lemma and a rooted C4-C5 signature amplifier, then develops an integral unimodular four-subdivision congruence preserving the determinant, adjacency cokernel, nonunit Smith factors, and nullity over every field. Two independent exact methods classify all 256 residue classes of three-cycle chains and agree row by row.

Key formulae

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

Document status

The manuscript is a non-peer-reviewed preprint. No journal acceptance or publication is claimed.

Main contributions

  • An infinite family of graphs in which the quantity at issue grows without bound, inside a class of simple graphs: drawable in the plane, at most three edges at any vertex, and cycles meeting in at most one point.
  • The technical piece that makes the construction possible: how to attach a block to a graph knowing in advance what effect it will have, and a block that amplifies that effect.
  • A way of splitting every edge into four while preserving the quantities that matter, and the exact classification of all 256 remaining cases.

Stated limitations

  • This is a preprint: no journal has reviewed it, and we claim no acceptance or publication.
  • This is Version 2 and it extends Version 1, which stays published with its own record: the earlier version is neither deleted nor corrected after the fact.
  • Revision 2.0-rev2 covers figures, exposition, certificates, documentation and checksums, as the deposit's own history states: no result changed. There is no separate tag for this revision in the public repository, so the source-code references describe Version 2.0.

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

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Date of this file
This file has existed since at least 1 August 2026. The proof sits in a public register we do not control, and anyone can check it.
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3e71e9f7c10e04e43ee40bd1ead03570c236c2c095b913f72c1eb8221c8ea7f1
SHA-256 · Zenodo reproducibility package
452eef35876f4d93741ddacd7c4942503d076e0a85b8d856455f47877817426c
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Timestamp receipt (.ots)
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Where to find this work elsewhere

Citation

BibTeX RIS
BibTeX 
@misc{Paone2026UnboundedSignatureLine,
  author = {Paone, Andrea},
  title = {Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles},
  month = aug,
  year = {2026},
  date = {2026-08-01},
  version = {2.0-rev2},
  doi = {10.5281/zenodo.21737348},
  url = {https://doi.org/10.5281/zenodo.21737348},
  note = {Preprint; not peer reviewed}
}
RIS 
TY  - UNPB
AU  - Paone, Andrea
TI  - Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles
PY  - 2026
DA  - 2026-08-01
DB  - Zenodo
ET  - 2.0-rev2
DO  - 10.5281/zenodo.21737348
UR  - https://doi.org/10.5281/zenodo.21737348
N1  - Preprint; not peer reviewed
KW  - line graph
KW  - graph signature
KW  - graph inertia
KW  - inertia indices
KW  - spectral graph theory
KW  - cactus graph
KW  - planar subcubic graph
KW  - rooted module
KW  - edge subdivision
KW  - integral unimodular congruence
KW  - Smith normal form
KW  - adjacency cokernel
KW  - exact residue classification
ER  -

Keywords