The MaxFlow command returns the optimal value for the max flow problem along with an optimal flow (as a Matrix).

The algorithm used is the Push-Relabel (Push-Preflow) algorithm of Goldberg et al. (see Introduction to Algorithms, Cormen, Leiserson, Rivest, 2nd edition). The complexity is $\mathrm{O}\left(n^{2}m\right)$ where n=|V| is the number of vertices of G and m=|E| is the number of edges.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[0\,1\,0\,4\,0\,0\right]\,\left[0\,0\,1\,0\,3\,0\right]\,\left[0\,1\,0\,0\,0\,1\right]\,\left[0\,0\,3\,0\,1\,0\right]\,\left[0\,0\,0\,1\,0\,4\right]\,\left[0\,0\,0\,0\,0\,0\right]\right]\right)$

$A≔\left[\begin{array}{cccccc}0& 1& 0& 4& 0& 0\\ 0& 0& 1& 0& 3& 0\\ 0& 1& 0& 0& 0& 1\\ 0& 0& 3& 0& 1& 0\\ 0& 0& 0& 1& 0& 4\\ 0& 0& 0& 0& 0& 0\end{array}\right]$

$N≔\mathrm{Digraph}\left(A\,\mathrm{weighted}\right)$

$N≔\mathrm{Graph\; 1:\; a\; directed\; weighted\; graph\; with\; 6\; vertices\; and\; 10\; arc(s)}$

$\mathrm{IsNetwork}\left(N\right)$

$\left\{1\right\},\left\{6\right\}$

$\mathrm{DrawNetwork}\left(N\right)$

$\mathrm{MaxFlow}\left(N\,1\,6\right)$

$4,\left[\begin{array}{cccccc}0& 1& 0& 3& 0& 0\\ 0& 0& 0& 0& 2& 0\\ 0& 1& 0& 0& 0& 1\\ 0& 0& 2& 0& 1& 0\\ 0& 0& 0& 0& 0& 3\\ 0& 0& 0& 0& 0& 0\end{array}\right]$