  The recently formulated theory of horizontal visibility graphs transforms
time series into graphs and allows the possibility of studying dynamical
systems through the characterization of their associated networks. This method
leads to a natural graph-theoretical description of nonlinear systems with
qualities in the spirit of symbolic dynamics. We support our claim via the case
study of the period-doubling and band-splitting attractor cascades that
characterize unimodal maps. We provide a universal analytical description of
this classic scenario in terms of the horizontal visibility graphs associated
with the dynamics within the attractors, that we call Feigenbaum graphs,
independent of map nonlinearity or other particulars. We derive exact results
for their degree distribution and related quantities, recast them in the
context of the renormalization group and find that its fixed points coincide
with those of network entropy optimization. Furthermore, we show that the
network entropy mimics the Lyapunov exponent of the map independently of its
sign, hinting at a Pesin-like relation equally valid out of chaos.
