Bizonotopal Graphical Algebras

TL;DR

This paper introduces bizonotopal algebras, a new family of graph-associated algebras doubling edges, revealing complex combinatorial properties and a novel deletion-contraction relation, expanding the algebraic understanding of graph invariants.

Contribution

The paper defines bizonotopal algebras, explores their properties, and shows they serve as complete invariants for graphs, differing from traditional zonotopal algebras.

Findings

Bizonotopal algebras are monomial and related to graphical parking functions.

Their Hilbert series satisfy a modified deletion-contraction relation.

External bizonotopal algebra uniquely determines the graph.

Abstract

Zonotopal algebras (external, central, and internal) of an undirected graph G introduced by Postnikov-Shapiro and Holtz-Ron, are finite-dimensional commutative graded algebras whose Hilbert series contain a wealth of combinatorial information about G. In this paper, we associate to G a new family of algebras, which we call bizonotopal, because their definition involves doubling the set of edges of G. These algebras are monomial and have intricate properties related, among other things, to the combinatorics of graphical parking functions and their polytopes. Unlike the case of usual zonotopal algebras, the Hilbert series of bizonotopal algebras are not specializations of the Tutte polynomial of G. Still, we show that in the external and central cases these Hilbert series satisfy a modified deletion-contraction relation. In addition, we prove that the external bizonotopal algebra is a complete graph invariant.