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

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 a suitable Abelian group A satisfying a coboundary condition (that the difference f − g is a flow over A of G) and a congruence condition (that an algebraic or combinatorial function of f and g, such as the sum of the size of the supports of f and g, satisfies some congruence condition). We prove several theorems of the form: a linear combination of sizes of parcels, with coefficients roots of unity, equals a multiple of an evaluation of the Tutte polynomial of M(G) at a point (u, v), usually with complex coordinates, satisfyingAn aim of the graph-theoretic results is to reformulate theorems, like the 4-color theorem, in probabilistic terms. The statements of these results involve differences in probabilities and their proofs are intricate, involving commutative algebra or finite Fourier analysis. Our aim in this paper is different: it is to put these results in context, as initial cases of infinite families of results about sizes of parcels constructed using flows of matrices over suitable Abelian groups.This paper is the second in a series. The earlier paper [13] is about the "parametric" theory. Using characters, we studied parcels defined by algebraic conditions using actual values of elements in the Abelian group A. With two exceptions, the present paper is about the "non-parametric" theory. Parcels are defined using congruence conditions which only use the data whether an element in A is zero or nonzero. We will use non-parametric parcels to obtain combinatorial interpretations of evaluations of rank generating polynomials on complex hyperbolas λx = q, where q is an integer greater than 1.(There seems to be only one interpretation of the evaluation of the rank generating polynomial at complex points in the literature, the interpretation by Jaeger [11] of the value of the Tutte polynomial at (e 2π ι/3 , e −2πι/3 ). Jaeger's interpretation will be discussed in Section 6.)We shall assume some knowledge of graph and matroid theory. See, for example, [1,12,16]. We recall briefly concepts and definitions central to this paper. Let G be a matrix with columns indexed by the set E and M(G) be the matroid on E defined by linear dependence of the column vectors.Let Γ (V , E) be a graph with a chosen orientation on the edges. The orientation gives a function sign(v, e): if e is not a loop, then sign(v, e) equals +1, −1, or 0, depending on whether the edge e is going into v, going out of v, or not incident at all on v. If e is a loop, then sign(v, e) is always 0. We will use two matrices defined by Γ . The first is the vertex-edge matrix, the |V | × |E| matrix H with rows indexed by V and columns indexed by E such that the ve-entry is sign(v, e). The matroid on E defined by the vertex-edge matrix is the cycle matroid of Γ . The second is the cycle-edge matrix. We think of a cycle c as a set of edges e 0 , e 1 , . . . , e t−1 such...