Lattice of integer flows and poset of strongly connected orientations

preprint2020arXivOpen access

Abstract

We show that the Voronoi cells of the lattice of integer flows of a finite connected graph GG in the quadratic vector space of real valued flows have the following very precise combinatorics: the face poset of a Voronoi cell is isomorphic to the poset of strongly connected orientations of subgraphs of GG. This confirms a conjecture of Caporaso and Viviani {Torelli Theorem For Graphs and Tropical Curves, Duke Math. J. 153(1) (2010), 129-171}.

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