The Lattice of Integral Flows and the Latticeof Integral Coboundaries on a Finite

The set of integral ows on a nite graph ? is naturally an integral lattice 1 (?) in the Euclidean space Ker((1) of harmonic real-valued functions on the edge set of ?. Various properties of ? (bipartite character, girth, complexity, separability) are shown to correspond to properties of 1 (?) (parity, minimal norm, determinant, decomposability). The dual lattice of 1 (?) is identiied to the integral cohomology H 1 (?; Z) in Ker((1). Analogous characterizations are shown to hold for the lattice of integral coboundaries and appropriate properties of the graph (Eulerian character, minimal bonds, complexity, separability). These lattices have a determinant group which plays for graphs the same role as Jacobians for closed Riemann surfaces. Analogs of Abel's theorem are established in the (much easier) setting of graphs and harmonic functions from graphs to abelian groups.