  We introduce and implement a method to compute stationary states of nonlinear
Schr\''odinger equations on metric graphs. Stationary states are obtained as
local minimizers of the nonlinear Schr\''odinger energy at fixed mass. Our
method is based on a normalized gradient flow for the energy (i.e. a gradient
flow projected on a fixed mass sphere) adapted to the context of nonlinear
quantum graphs. We first prove that, at the continuous level, the normalized
gradient flow is well-posed, mass-preserving, energy diminishing and converges
(at least locally) towards stationary states. We then establish the link
between the continuous flow and its discretized version. We conclude by
conducting a series of numerical experiments in model situations showing the
good performance of the discrete flow to compute stationary states. Further
experiments as well as detailed explanation of our numerical algorithm are
given in a companion paper.
