  The structure of pipe networks minimizing the total energy dissipation rate
is studied analytically. Among all the possible pipe networks that can be built
with a given total pipe volume (or pipe lateral surface area), the network
which minimizes the dissipation rate is shown to be loopless. Furthermore, such
an optimal network is shown to contain at most N-2 nodes in addition to the N
sources plus sinks that it connects. These results are valid whether the
possible locations for the additional nodes are chosen freely or from a set of
nodes (such as points of a grid). Applications of these results to various
physical situations and to the efficient computation of optimal pipe networks
are also discussed.
