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.
Other
Version note
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. Version 2.0 adds the unbounded family, rooted-module transfer theorem, integral four-subdivision congruence, and exact residue classification.
Status: preprint, not peer reviewed.
Files
unbounded_signature_line_graphs.pdf
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.