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In The curl of graphs and networks (Gustafson and Haray, 1984) it is claimed to be shown that any digraph $G$ can be decomposed as the sum of three graphs $U_1 + U_2 + U_3$, where $U_1$ is divergence-free, $U_2$ is curl-free and $U_3$ is both curl and divergence free.

However I am having a hard time understanding the proposed definitions for curl and divergence, and the decomposition.

It would be immensely helpful if somebody could help me decipher the paper by providing an example of a digraph decomposition.

For example, how could one decompose the following weighted digraph?

enter image description here

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  • $\begingroup$ isn't this fully solenoidal (divergence-free) ? $\endgroup$ Feb 10, 2020 at 14:44
  • $\begingroup$ @CarloBeenakker is it? How I was undertstanding it (and this may betray how little I do understand), in a solenoidal graph we would have that each node satifies that the inflow equals the outflow $\endgroup$ Feb 10, 2020 at 14:49

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