Critical Groups Detect Homology Actions of Simple Graphs
Manuscript 19 October 2026 · Online 19 October 2026
Abstract
For every finite connected simple graph, we prove that the natural actions of its automorphism group on the finite critical group and on real first homology have the same kernel. The proof uses the cut-flow presentation and the absence of integral flows of squared norm at most two, with a separate coordinate argument for reversed edges. Combining this equality with known homological faithfulness yields a faithful critical-group action for every connected simple graph of minimum degree at least two other than a cycle. In particular, semiregularity is unnecessary for simple three-connected graphs. Parallel-edge and pendant-tree examples explain the hypotheses. This is a complete scoped theorem within AIM-COMBINATORICS-0131, not closure of the open-ended AIM research suggestion. The preprint is AI-assisted, self-audited and unrefereed.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Complete kernel theorem
- Categories
- math.CO · math.GR
- Manuscript
- 19 October 2026
- Online release
- 19 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Critical Groups Detect Homology Actions of Simple Graphs,” EulerSolve Research Papers, AIM-COMBINATORICS-0131, 2026. https://doi.org/10.5281/zenodo.23471711.
BibTeX
@misc{ferudun2026criticalhomology,
author = {Ferudun, Alper},
title = {Critical Groups Detect Homology Actions of Simple Graphs},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-combinatorics-0131/},
doi = {10.5281/zenodo.23471711},
note = {AIM-COMBINATORICS-0131; unrefereed preprint}
}