Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles
Description
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. Version 1 exhibited a connected simple counterexample with line-graph inertia (9, 0, 7). Version 2 proves that the failure is unbounded even for connected simple planar subcubic cactus graphs.
It establishes a rooted-module attachment lemma and an explicit rooted C4-C5 signature amplifier, yielding a family whose line-graph signature grows by one at each attachment. It also proves an arbitrary-edge integral unimodular four-subdivision congruence preserving determinant, adjacency cokernel, nonunit Smith factors, and nullity over every field.
Finally, two independent exact methods classify all 256 residue classes of three-cycle chains and agree row by row.
Notes
Version 2.0-rev2 improves the reader-facing presentation and the reproducibility documentation. It adds schematic figures for the seed, rooted attachment, alternating family, and four-subdivision operation; makes the seed and amplifier certificates directly checkable from the manuscript; and rewrites the abstract, reproducibility statement, and discussion in ordinary scholarly prose.
Version 2.0-rev1 is a bibliographic revision of Version 2.0, first published on 24 July 2026. It adds a targeted citation and related-work comparison with Francis and Uptain, arXiv:2607.22874, and records the revision lineage. No theorem, proof, equation, exact certificate, computational script, result, or numerical statement has changed.
Version 2.0 substantially expands and supersedes Version 1.0 as the current research version, while preserving Version 1.0 as the historical finite-counterexample record. It adds the unbounded family, rooted-module transfer theorem, integral four-subdivision congruence, and exact residue classification.
Status: Preprint; not peer reviewed.
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unbounded_signature_line_graphs_v2.0-rev2.pdf
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Additional details
Related works
- Cites
- Publication: 10.1016/j.disc.2025.114953 (DOI)
Software
- Repository URL
- https://github.com/Andrew3000s/line-graph-inertia-public
- Programming language
- Python
- Development Status
- Active
References
- Akbari, S., Elphick, C., Kumar, H., Pragada, S., and Tang, Q. "A new conjecture on the inertia of graphs." Discrete Mathematics 349(4) (2026), Article 114953. https://doi.org/10.1016/j.disc.2025.114953.
- Chen, H., and Li, J. "Counterexamples to a conjecture on graph inertia." arXiv:2605.07196v1, 2026. https://doi.org/10.48550/arXiv.2605.07196.