  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 repulsion has a maximum at a critical
$N$-dependent value of the on-site Hubbard $U$. 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. We are aware of no previous 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.
