The Negami Polynomial and Broken-Circuit Stanley--Reisner Rings
Abstract
We summarize the classical passage from the chromatic specialization of Negami's polynomial to the Hilbert series of a broken-circuit Stanley--Reisner ring, and record formulas for several graph families with their connectivity hypotheses made explicit. Our main construction in this revision is the associated graded algebra of the squarefree edge algebra under the graphic-matroid rank filtration, for an arbitrary finite loopless graph. Its basis is indexed by all edge subsets, and its multiplication is governed by disjointness and rank additivity. Its bigraded Hilbert polynomial recovers the full three-variable Negami polynomial. We prove that its socle has a basis indexed by edge subsets of full matroid rank; for a connected graph, these are precisely the connected spanning subgraphs. A vertex-degree refinement counts connected regular spanning subgraphs. This finite-dimensional algebra is distinct from a broken-circuit Stanley--Reisner ring. We distinguish these elementary structural results from the unresolved question of whether abstract algebra isomorphism can distinguish graphs with the same Negami polynomial; no priority claim is made for the construction or its Hilbert identity.
Related
- Local Shellings of Broken-Circuit Complexes via Boundary States and the Chromatic Specialization of the Negami Polynomial
- Stanley-Reisner Theory in Mixed Characteristic
- Cube-freeness and a square formula for canonical trace ideals of Stanley--Reisner rings
- Characterization of Stanley-Reisner varieties by their automorphism group
- Powers of Alexander Duals of Stanley-Reisner Ideals of Chessboard Complexes
BibTeX
@misc{mizukai2026negami,
title = {{The Negami Polynomial and Broken-Circuit Stanley--Reisner Rings}},
author = {Iwao Mizukai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.05936},
eprint = {2609.05936},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05936},
}
RIS
TY - UNPB
TI - The Negami Polynomial and Broken-Circuit Stanley--Reisner Rings
AU - Iwao Mizukai
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2609.05936
UR - https://arxiv.org/abs/2609.05936
AB - We summarize the classical passage from the chromatic specialization of Negami's polynomial to the Hilbert series of a broken-circuit Stanley--Reisner ring, and record formulas for several graph families with their connectivity hypotheses made explicit. Our main construction in this revision is the associated graded algebra of the squarefree edge algebra under the graphic-matroid rank filtration, for an arbitrary finite loopless graph. Its basis is indexed by all edge subsets, and its multiplication is governed by disjointness and rank additivity. Its bigraded Hilbert polynomial recovers the full three-variable Negami polynomial. We prove that its socle has a basis indexed by edge subsets of full matroid rank; for a connected graph, these are precisely the connected spanning subgraphs. A vertex-degree refinement counts connected regular spanning subgraphs. This finite-dimensional algebra is distinct from a broken-circuit Stanley--Reisner ring. We distinguish these elementary structural results from the unresolved question of whether abstract algebra isomorphism can distinguish graphs with the same Negami polynomial; no priority claim is made for the construction or its Hilbert identity.
ER -