  The Friedel sum rule is extended to deal with topological defects for the
case of a graphene cone in the presence of an external Coulomb charge. The
dependence in the way the number of states change due to both the topological
defect as well as the Coulomb charge are studied. Our analysis addresses both
the cases of a subcritical as well as a supercritical value of the Coulomb
charge. We also discuss the experimental implications of introducing a
self-adjoint extension of the system Hamiltonian. We argue that the boundary
conditions following from the self-adjoint extension encode the effect of short
range interactions present in the system.
