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

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

2010-07-02

arxiv.org/abs/1007.0279v2

Mathematics

Combinatorics

Scientific paper

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 $(\lambda - 1)(x-1) = |A|.$

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