  We investigate the way that the degenerate manifold of midgap edge states in
quasicircular graphene quantum dots with zig-zag boundaries supports, under
free-magnetic-field conditions, strongly correlated many-body behavior
analogous to the fractional quantum Hall effect (FQHE), familiar from the case
of semiconductor heterostructures in high magnetic fields. Systematic
exact-diagonalization (EXD) numerical studies are presented for the first time
for 5 <= N <= 8 fully spin-polarized electrons and for total angular momenta in
the range of N(N-1)/2 <= L <= 150. We present a derivation of a
rotating-electron-molecule (REM) type wave function based on the methodology
introduced earlier [C. Yannouleas and U. Landman, Phys. Rev. B 66, 115315
(2002)] in the context of the FQHE in two-dimensional semiconductor quantum
dots. The EXD wave functions are compared with FQHE trial functions of the
Laughlin and the derived REM types. It is found that a variational extension of
the REM offers a better description for all fractional fillings compared with
that of the Laughlin functions (including total energies and overlaps), a fact
that reflects the strong azimuthal localization of the edge electrons. In
contrast with the multiring arrangements of electrons in circular semiconductor
quantum dots, the graphene REMs exhibit in all instances a single (0,N)
polygonal-ring molecular (crystalline) structure, with all the electrons
localized on the edge. Disruptions in the zig-zag boundary condition along the
circular edge act effectively as impurities that pin the electron molecule,
yielding single-particle densities with broken rotational symmetry that portray
directly the azimuthal localization of the edge electrons.
