  We present a complete analytical solution of the mean-field Hubbard model of
undoped and doped graphene rectangulenes. These are non-chiral ribbons of
arbitrary length and width, whose dimensions range from simple short acene
molecules all the way up to the bulk limit. We rewrite the Hubbard model in the
basis of bulk and edge non-interacting eigen-states, and provide explicit
expressions for the Coulomb matrix elements. We present a general mean-field
decoupling of the Hamiltonian, and discuss in detail the paramagnetic,
ferromagnetic and antiferromagnetic mean-field solutions. We calculate the
eigen-energies, occupations, spin densities and addition energies of
rectangulenes with lengths and widths ranging from a nanometer to several
hundreds of them. We rewrite the exact mean-field tight-binding Hamiltonian
back in the site-occupation basis, that can be used to model electronic,
thermo-electric, transport and optical properties of experimental-size graphene
flakes.
