  An energy gap can be opened in the electronic spectrum of graphene by lifting
its sublattice symmetry. In bilayers, it is possible to open gaps as large as
0.2 eV. However, these gaps rarely lead to a highly insulating state expected
for such semiconductors at low temperatures. This long-standing puzzle is
usually explained by charge inhomogeneity. Here we investigate spatial
distributions of proximity-induced superconducting currents in gapped graphene
and, also, compare measurements in the Hall bar and Corbino geometries in the
normal state. By gradually opening the gap in bilayer graphene, we find that
the supercurrent at the charge neutrality point changes from uniform to such
that it propagates along narrow stripes near graphene edges. Similar stripes
are found in gapped monolayers. These observations are corroborated by using
the "edgeless" Corbino geometry in which case resistivity at the neutrality
point increases exponentially with increasing the gap, as expected for an
ordinary semiconductor. This is in contrast to the Hall bar geometry where
resistivity measured under similar conditions saturates to values of only about
a few resistance quanta. We attribute the metallic-like edge conductance to a
nontrivial topology of gapped Dirac spectra.
