  We analyze the screening of an external Coulomb charge in gapless graphene
cone, which is taken as a prototype of a topological defect. In the subcritical
regime, the induced charge is calculated using both the Green's function and
the Friedel sum rule. The dependence of the polarization charge on the Coulomb
strength obtained from the Green's function clearly shows the effect of the
conical defect and indicates that the critical charge itself depends on the
sample topology. Similar analysis using the Friedel sum rule indicates that the
two results agree for low values of the Coulomb charge but differ for the
higher strengths, especially in the presence of the conical defect. For a given
subcritical charge, the transport cross-section has a higher value in the
presence of the conical defect. In the supercritical regime we show that the
coefficient of the power law tail of polarization charge density can be
expressed as a summation of functions which vary log periodically with the
distance from the Coulomb impurity. The period of variation depends on the
conical defect. In the presence of the conical defect, the Fano resonances
begin to appear in the transport cross-section for a lower value of the Coulomb
charge. For both sub and supercritical regime we derive the dependence of LDOS
on the conical defect. The effects of generalized boundary condition on the
physical observables are also discussed.
