  We comprehensively show the topological structure--order of zeros felt by an
electron at the positions of other electrons --for the enigmatic filling factor
5/2 at the "Pfaffian-shift" in a spherical geometry. A set of linearly
independent antisymmetric functions that are constructed from the possible
graphs preserving this topological structure provides complete basis for
determining an accurate ground state wave function for any interaction, in
particular the Coulomb interaction. One of the graphs describes $Z_2$
para-fermion equivalently the Pfaffian wave function, but some of the other
graphs also contribute to form the exact ground state for the Coulomb
interaction. We further show that our wave function for the Coulomb interaction
supports clustering up to half of the composite bosons, signifying a
strong-coupling regime.
