The Loopy Polynomial: from Tutte's Universal $V$-Function to Bizonotopal Geometry
We study the loopy polynomial L_G, a multivariate graph invariant arising from bizonotopal graph algebras and defined by a deletion-loopy-contraction recursion, in which the contracted edge becomes a loop. We show that L_G contains the Tutte polynomial and admits a similar spanning-forest activity expansion. It also determines Stanley's chromatic symmetric function, the degree sequence, the induced edge count profile, and the independence polynomial for loopless graphs, and the clique and matching polynomials for simple graphs. Separating the size and the external activity of each forest component leads to a refined loopy polynomial, which we show to be equivalent to the extended U-polynomial of Noble and Welsh and to the extended polychromate. Different specializations of this common refinement give the ordinary U-polynomial, Tutte's universal V-function, and Stanley's Tutte symmetric function, placing these invariants into a single framework. We conjecture that L_G and the U-polynomial have the same distinguishing power on simple graphs, and verify this for all graphs on at most 11 vertices. Simplicity is essential: we found two loopless multigraphs with equal U-polynomials but distinct loopy polynomials. They also have distinct extended U-polynomials, so the ordinary U-polynomial does not determine the extended one on loopless multigraphs. This solves an open problem by Merino and Noble. For the score polytope P_G of the external bizonotopal algebra, loopy deletion-contraction lifts from the lattice-point enumerator to the polytope itself. This gives forest-indexed geometric parking complexes whose lattice points partition those of P_G, and which are piecewise-linearly parametrized by products of intervals whose lengths are the component weights in the forest expansion of L_G.