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The Lattice Of Integral Flows And The Lattice Of Integral Cuts On A Finite Graph
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BibTeX
@MISC{Bacher_thelattice,
author = {Roland Bacher and Pierre de la Harpe and Tatiana Nagnibeda},
title = {The Lattice Of Integral Flows And The Lattice Of Integral Cuts On A Finite Graph},
year = {}
}
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Abstract
. The set of integral flows on a finite graph \Gamma is naturally an integral lattice 1 (\Gamma) in the Euclidean space 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 1 (\Gamma) (parity, minimal norm, determinant, decomposability). The dual lattice of 1 (\Gamma) is identified to the integral cohomology H 1 (\Gamma; Z) in Ker(\Delta 1 ). Analogous characterizations are shown to hold for the lattice of integral cuts 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. Partial analogs of Abel's theorem are established in the (much easier) setting of graphs and harmonic functions from graphs to abelian groups. 1. Introduction: The lattices 1 (\Gamma) and N 1...
Keyphrases
integral flow integral cut finite graph eulerian character abelian group partial analog euclidean space ker finite graph gamma various property integral cohomology dual lattice determinant group harmonic function bipartite character edge set harmonic real-valued function analogous characterization integral lattice minimal norm appropriate property minimal bond closed riemann surface
