  We provide new results on the structure of optimal transportation networks
obtained as minimizers of an energy cost functional consisting of a kinetic
(pumping) and material (metabolic) cost terms, constrained by a local mass
conservation law. In particular, we prove that every tree (i.e., graph without
loops) represents a local minimizer of the energy with concave metabolic cost.
For the linear metabolic cost, we prove that the set of minimizers contains a
loop-free structure. Moreover, we enrich the energy functional such that it
accounts also for robustness of the network, measured in terms of the Fiedler
number of the graph with edge weights given by their conductivities. We examine
fundamental properties of the modified functional, in particular, its convexity
and differentiability. We provide analytical insights into the new model by
considering two simple examples. Subsequently, we employ the projected
subgradient method to find global minimizers of the modified functional
numerically. We then present two numerical examples, illustrating how the
optimal graph's structure and energy expenditure depend on the required
robustness of the network.
