  In this paper we demonstrate, using a couple of variants of a two-strand
ladder network that, a quasiperiodic Aubry-Andr\'e-Harper (AAH) modulation
applied to the vertical strands, mimicking a deterministic distortion in the
network, can give rise to certain exotic features in the electronic spectrum of
such systems. While, for the simplest ladder network all the eigenstates become
localized as the modulation strength crosses a threshold, for the second
variant, modelling an ultrathin graphene nano-ribbon, the central part of the
energy spectrum remains populated by extended wavefunctions. The multifractal
character in the energy spectrum is observed for both these networks close to
the critical values of the modulation. We substantiate our findings also by
studying the quantum dynamics of a wave packet on such decorated lattices.
Interestingly, while the mean square displacement (MSD) changes in the usual
manner in a pure two-strand ladder network as the modulation strength varies,
for the ultrathin graphene nanoribbon the temporal behaviour of the MSD remains
unaltered only up to a strong modulation strength. This, we argue, is due to
the extendedness of the wavefunction at the central part of the energy
spectrum. Other measurements like the return probability, temporal
autocorrelation function, the time dependence of the inverse participation
ratio, and the information entropy are calculated for both networks with
different modulation strengths and corroborate our analytical findings.
