  As a prototype model of antiferromagnetism, we propose a repulsive Hubbard
Hamiltonian defined on a graph $\L={\cal A}\cup{\cal B}$ with ${\cal A}\cap
{\cal B}=\emptyset$ and bonds connecting any element of ${\cal A}$ with all the
elements of ${\cal B}$. Since all the hopping matrix elements associated with
each bond are equal, the model is invariant under an arbitrary permutation of
the ${\cal A}$-sites and/or of the ${\cal B}$-sites. This is the Hubbard model
defined on the so called $(N_{A},N_{B})$-complete-bipartite graph, $N_{A}$
($N_{B}$) being the number of elements in ${\cal A}$ (${\cal B}$). In this
paper we analytically find the {\it exact} ground state for $N_{A}=N_{B}=N$ at
half filling for any $N$; the ground state expectation value of the repulsion
term has a maximum at a critical $N$-dependent value of the on-site Hubbard $U$
and then drops like 1/U for large $U$. The wave function and the energy of the
unique, singlet ground state assume a particularly elegant form for $N \ra
\inf$. We also calculate the spin-spin correlation function and show that the
ground state exhibits an antiferromagnetic order for any non-zero $U$ even in
the thermodynamic limit. This is the first explicit analytic example of an
antiferromagnetic ground state in a Hubbard-like model of itinerant electrons.
The kinetic term induces non-trivial correlations among the particles and an
antiparallel spin configuration in the two sublattices comes to be
energetically favoured at zero Temperature. On the other hand, if the
thermodynamic limit is taken and then zero Temperature is approached, a
paramagnetic behavior results. The thermodynamic limit does not commute with
the zero-Temperature limit, and this fact can be made explicit by the analytic
solutions.
