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The lattice of integral flows and the lattice of integral cuts on a finite graph. (English) Zbl 0891.05062
Authors’ abstract: The set of integral flows on a finite graph \(\Gamma\) is naturally an integral lattice \(\Lambda^1 (\Gamma)\) in the Euclidean space \(\text{Ker} (\Delta_1)\) of harmonic real-valued functions on the edge set of \(\Gamma\). Various properties of \(\Gamma\) (bipartite character, girth, complexity, separability) are shown to correspond to properties of \(\Lambda^1 (\Gamma)\) (parity, minimal norm, determinant, decomposability). The dual lattice of \(\Lambda^1 (\Gamma)\) is identified to the integral cohomology \(H^1(\Gamma, \mathbb{Z})\) in \(\text{Ker} (\Delta_1)\). 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.

MSC:
05C99 Graph theory
11E39 Bilinear and Hermitian forms
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