Congruence conditions, parcels, and Tutte polynomials of graphs and matroids

TL;DR

This paper explores the relationship between parcels, coboundary and congruence conditions, and Tutte polynomial evaluations of graphs and matroids, revealing new algebraic identities involving roots of unity.

Contribution

It introduces a novel framework linking parcels with Tutte polynomial evaluations via algebraic identities involving roots of unity.

Findings

Linear combinations of parcel sizes equal Tutte polynomial evaluations

Identifies conditions under which these identities hold on complex hyperbola

Connects algebraic parcel structures with combinatorial invariants

Abstract

Let $G$ be a matrix and $M(G)$ be the matroid defined by linear dependence on the set $E$ of column vectors of $G.$ Roughly speaking, a parcel is a subset of pairs $(f,g)$ of functions defined on $E$ to an Abelian group $A$ satisfying a coboundary condition (that $f-g$ is a flow over $A$ relative to $G$) and a congruence condition (that the size of the supports of $f$ and $g$ satisfy some congruence condition modulo an integer). We prove several theorems of the form: a linear combination of sizes of parcels, with coefficients roots of unity, equals an evaluation of the Tutte polynomial of $M(G)$ at a point $(\lambda-1,x-1)$ on the complex hyperbola $(\lambda - 1)(x-1) = |A|.$