The present paper is about a concept from mathematical analysis, namely the Helmholtz decomposition of vector fields that can be translated into a similar concept for certain directed graphs, also called the Helmholtz decomposition.
The classical Helmholtz decomposition decomposes a vector field into two components: one part is defined by a potential (and hence is rotation free) and the other part is derived from a vector potential (and hence is divergence free). A similar decomposition is possible for a digraphs with valuations defined on its arcs. It is possible to decompose the valuation in a similar way: one part has the property that the sum of the values associated with the arcs on a cycle is zero; the other part has the property that the sum of the values of the ingoing arcs at a node is equal to the sum of the values of the outgoing arcs, at that node.
