The lattice of integral flows and the lattice of integral cuts on a finite graph

Abstract

The set of integral flows on a finite graph Γ is naturally an integral lattice ∧1 (Γ) in the Euclidean space Ker(Δ1) of harmonic real-valued functions on the edge set of Γ. Various properties of Γ (bipartite character, girth, complexity, separability) are shown to correspond to properties of ∧1(Γ) (parity, minimal norm, determinant, decomposability). The dual lattice of ∧1(Γ) is identified to the integral cohomology H1(Γ,ℤ) in Ker(Δ1). Analogous characterizations are shown to hold for the lattice of integral cuts and appropriate properties of the graph (Eulerian character, edge connectivity, complexity, separability). These lattices have a determinant group which plays for graphs the same role as Jacobians for closed Riemann surfaces. It is then harmonic functions on a graph (with values in an abelian group) which take place of holomorphic mappings.