  Correlated charge inhomogeneity breaks the electron-hole symmetry in
two-dimensional (2D) bilayer heterostructures which is responsible for non-zero
drag appearing at the charge neutrality point. Here we report Coulomb drag in
novel drag systems consisting of a two-dimensional graphene and a one
dimensional (1D) InAs nanowire (NW) heterostructure exhibiting distinct results
from 2D-2D heterostructures. For monolayer graphene (MLG)-NW heterostructures,
we observe an unconventional drag resistance peak near the Dirac point due to
the correlated inter-layer charge puddles. The drag signal decreases
monotonically with temperature ($\sim T^{-2}$) and with the carrier density of
NW ($\sim n_{N}^{-4}$), but increases rapidly with magnetic field ($\sim
B^{2}$). These anomalous responses, together with the mismatched thermal
conductivities of graphene and NWs, establish the energy drag as the
responsible mechanism of Coulomb drag in MLG-NW devices. In contrast, for
bilayer graphene (BLG)-NW devices the drag resistance reverses sign across the
Dirac point and the magnitude of the drag signal decreases with the carrier
density of the NW ($\sim n_{N}^{-1.5}$), consistent with the momentum drag but
remains almost constant with magnetic field and temperature. This deviation
from the expected $T^2$ arises due to the shift of the drag maximum on graphene
carrier density. We also show that the Onsager reciprocity relation is observed
for the BLG-NW devices but not for the MLG-NW devices. These Coulomb drag
measurements in dimensionally mismatched (2D-1D) systems, hitherto not
reported, will pave the future realization of correlated condensate states in
novel systems.
