The Loopy Polynomial: from Tutte's Universal V-Function to Bizonotopal Geometry
Abstract
We study the loopy polynomial L_G, a multivariate graph invariant arising from bizonotopal graphical algebras and defined by deletion and loopy contraction, in which the contracted edge is retained as a loop. It determines the Tutte polynomial and admits a spanning-forest expansion recording external activity componentwise. It also determines Stanley's chromatic symmetric function for every graph; the degree sequence, induced edge count profile, and independence polynomial for loopless graphs; and the clique and matching polynomials for simple graphs. Separating component size from external activity gives a refinement that is equivalent to the extended U-polynomial and the extended polychromate. The ordinary U-polynomial, Tutte's universal V-function, and Stanley's Tutte symmetric function are specializations of this refinement. We conjecture that L_G and U_G have the same distinguishing power on simple graphs, and verify this through eleven vertices. We exhibit five-vertex loopless multigraphs with equal U-polynomials but distinct loopy and extended U-polynomials, solving a problem of Merino and Noble. We also disprove Loebl's conjecture for chordal graphs and its restriction to split graphs, with counterexamples on eleven and fourteen vertices, respectively. For the score polytope P_G of the external bizonotopal algebra, we give a geometric realization of loopy deletion-contraction. The resulting forest-indexed geometric parking complexes partition the lattice points of P_G and are piecewise integral-affinely parametrized by boxes whose interval lengths are the component indices in the forest expansion.
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BibTeX
@misc{kirillov2026loopy,
title = {{The Loopy Polynomial: from Tutte's Universal $V$-Function to Bizonotopal Geometry}},
author = {Anatol Kirillov and Gleb Nenashev and Boris Shapiro and Arkady Vaintrob},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.07728},
eprint = {2609.07728},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07728},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Loopy Polynomial: from Tutte's Universal V-Function to Bizonotopal Geometry
AU - Anatol Kirillov
AU - Gleb Nenashev
AU - Boris Shapiro
AU - Arkady Vaintrob
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2609.07728
UR - https://arxiv.org/abs/2609.07728
AB - We study the loopy polynomial L_G, a multivariate graph invariant arising from bizonotopal graphical algebras and defined by deletion and loopy contraction, in which the contracted edge is retained as a loop. It determines the Tutte polynomial and admits a spanning-forest expansion recording external activity componentwise. It also determines Stanley's chromatic symmetric function for every graph; the degree sequence, induced edge count profile, and independence polynomial for loopless graphs; and the clique and matching polynomials for simple graphs. Separating component size from external activity gives a refinement that is equivalent to the extended U-polynomial and the extended polychromate. The ordinary U-polynomial, Tutte's universal V-function, and Stanley's Tutte symmetric function are specializations of this refinement. We conjecture that L_G and U_G have the same distinguishing power on simple graphs, and verify this through eleven vertices. We exhibit five-vertex loopless multigraphs with equal U-polynomials but distinct loopy and extended U-polynomials, solving a problem of Merino and Noble. We also disprove Loebl's conjecture for chordal graphs and its restriction to split graphs, with counterexamples on eleven and fourteen vertices, respectively. For the score polytope P_G of the external bizonotopal algebra, we give a geometric realization of loopy deletion-contraction. The resulting forest-indexed geometric parking complexes partition the lattice points of P_G and are piecewise integral-affinely parametrized by boxes whose interval lengths are the component indices in the forest expansion.
ER -