Orientations, lattice polytopes, and group arrangements III: Cartesian product arrangements and applications to the Tutte type polynomials of graphs

TL;DR

This paper introduces new polynomial invariants for graphs based on product arrangements and tension-flows, extending the combinatorial understanding of Tutte polynomials and their special cases.

Contribution

It develops a theory of product arrangements and multivariable characteristic polynomials, linking them to Tutte and Whitney polynomials, and explores tension-flow types in graphs.

Findings

Defined three tension-flow types: elliptic, parabolic, hyperbolic.

Established reciprocity laws for weighted polynomials.

Unified Tutte polynomial as a special case of dual parabolic polynomials.

Abstract

A common generalization for the chromatic polynomial and the flow polynomial of a graph $G$ is the Tutte polynomial $T(G;x,y)$. The combinatorial meaning for the coefficients of $T$ was discovered by Tutte at the beginning of its definition. However, for a long time the combinatorial meaning for the values of $T$ is missing, except for a few values such as $T(G;i,j)$, where $1\leq i,j\leq 2$, until recently for $T(G;1,0)$ and $T(G;0,1)$. In this third one of a series of papers, we introduce product valuations, cartesian product arrangements, and multivariable characteristic polynomials, and apply the theory of product arrangement to the tension-flow group associated with graphs. Three types of tension-flows are studied in details: elliptic, parabolic, and hyperbolic; each type produces a two-variable polynomial for graphs. Weighted polynomials are introduced and their reciprocity laws are obtained. The dual versions for the parabolic case turns out to include Whitney's rank generating polynomial and the Tutte polynomial as special cases. The product arrangement part is of interest for its own right. The application part to graphs can be modified to matroids.