FIG. 2. Two examples of growing ladder graphs: in (a) all the borders are connected (remember that, following the definition we use here, a border divides a graph into exactly two subgraphs, each connected ), and as a consequence the graph will magnetize. In (b) too the borders are connected, and even if the growth in width of the ladder is very slow (logarithmic) the same result holds. is proved that h ( n ) ≥ log n is both a sufficient and a necessary condition for spontaneous magnetization. To give a more intuitive interpretation of the theorem we proved, we can proceed as follows: if the number of borders grows less than exponentially, we can argue that all of these borders will contain a number of spins increasing slowly with the length of the border; as a consequence, the formation of large clusters of spins in a magnetized graph will be energetically unfavoured, so that the latter will result a stable state. On the other hand, if µ $^{i}$( b ) grows very fast with b , we expect that some of the borders will be far from the vertex i , so that more and more vertices will be enclosed in short (low b ) borders; this in turn means that large clusters of spins can be flipped spending a small amount of energy, so that a magnetized graph may be unstable with regard to thermal fluctuations. The condition of our theorem is a strong one, in that it investigates a global property of the graph. For this reason it can not be a necessary condition for achieving spontaneous magnetization: if a graph has a part, which has zero measure in the thermodynamic limit, DISCUSSION 13