  The transport properties of nematic aerogels, which consist of highly
oriented Al$_2$O$_3\cdot$SiO$_2$ nanofibers coated with a graphene shell with a
large number of defects, are studied. The temperature dependences of the
electrical resistivity in the range of 9-40K strictly follow the formula
derived to describe the variable range hopping (VRH) conductivity, in which
exponent $\alpha$ changes from 0.4 to 0.9 when the number of layers in the
graphene shell decreases from 4-6 to 1-2. The dependence of $\alpha$ on the
shell thickness can be explained by a simultaneous change in the dimensionality
of hopping transport and the character of the energy dependence of the density
of localized states near the Fermi level. The fact that $\alpha$ approaches
unity at the minimum graphene shell thickness indicates a gradual transition
from VRH transport to nearest neighbor hopping (NNH) transport. The
magnetoresistance measured at T = 4.2 K is negative, increases significantly
with decreasing graphene shell thickness, and is approximated by a formula for
the case of weak localization with a good accuracy. The phase coherence lengths
are in a reasonable relation with the graphene grain sizes. The conducting
aerogels under study complement the well-known set of materials that exhibit
hopping electron transport at low temperatures, which is characteristic of
media with strong carrier localization, and also a negative magnetoresistance,
which usually manifests itself under weak localization conditions.
