  Time series are proficiently converted into graphs via the horizontal
visibility (HV) algorithm, which prompts interest in its capability for
capturing the nature of different classes of series in a network context. We
have recently shown [1] that dynamical systems can be studied from a novel
perspective via the use of this method. Specifically, the period-doubling and
band-splitting attractor cascades that characterize unimodal maps transform
into families of graphs that turn out to be independent of map nonlinearity or
other particulars. Here we provide an in depth description of the HV treatment
of the Feigenbaum scenario, together with analytical derivations that relate to
the degree distributions, mean distances, clustering coefficients, etc.,
associated to the bifurcation cascades and their accumulation points. We
describe how the resultant families of graphs can be framed into a
renormalization group scheme in which fixed-point graphs reveal their scaling
properties. These fixed points are then re-derived from an entropy optimization
process defined for the graph sets, confirming a suggested connection between
renormalization group and entropy optimization. Finally, we provide analytical
and numerical results for the graph entropy and show that it emulates the
Lyapunov exponent of the map independently of its sign.
