The problem of computing flows in combined dendritic and looped networks of channels is shown to be closely related to the hydraulic properties of the network. This relation is expressed here in the simplest of graph-theoretical terms in such a way as to provide a completely general, unconditionally stable, accurate and fast method of solution. In particular, once any such model is instantiated, its running time increases only linearly with the number of grid points used, regardless of the complexity of the network. The algorithms, which are elaborated here for the example of sub-critical flow conditions using computer-algebraic procedures, can be easily applied to urban-drainage and irrigation networks in a physically transparent way. They thus lend themselves particularly well to mixed logical and numerical optimisation in real-time control applications.