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

Abstract

Let GG be a matrix and M(G)M(G) be the matroid defined by linear dependence on the set EE of column vectors of G.G. Roughly speaking, a parcel is a subset of pairs (f,g)(f,g) of functions defined on EE to an Abelian group AA satisfying a coboundary condition (that fgf-g is a flow over AA relative to GG) and a congruence condition (that the size of the supports of ff and gg 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)M(G) at a point (λ1,x1)(\lambda-1,x-1) on the complex hyperbola $(\lambda - 1)(x-1) = |A|.

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