  In this paper, we study combinatorial properties of stable curves. To the
dual graph of any nodal curve, it is naturally associated a group, which is the
group of components of the N\'eron model of the generalized Jacobian of the
curve. We study the order of this group, called the complexity. In particular,
we provide a partial characterization of the stable curves having maximal
complexity, and we provide an upper bound, depending only on the genus $g$ of
the curve, on the maximal complexity of stable curves; this bound is
asymptotically sharp for $g\gg 0$. Eventually, we state some conjectures on the
behavior of stable curves with maximal complexity, and prove partial results in
this direction.
