validity of the theorem, all the weakly separable [22] graphs, which do not magnetize, belong to this category. To further our understanding of the result, we need to present a new parameter. Given a subgraph A ⊂ G , we define its external boundary ∂A as the set of points in G \ A that have a bond to a point in A ; denoting the number of vertices in A as | A | we now present the isoperimetric dimension d$_{iso}$ as the minimum d such that ∂A ≥ C · | A | d − 1 d . The largest set of points encompassable with b links is thus smaller than b d iso d iso − 1 ; since these points are connected, b d iso d iso − 1 is also the maximum radius of a set including i with a border b , so the set of reachable points, V ( b ) ⊂ P , has a cardinality | V ( b ) | ≤ b$^{d}$ iso d frac d iso − 1 . Given a point p and for each border B , consider ¯ B , the collection of vertices contributing to B which are on the inside of B with respect to p . As the two are in biunivocal relation once p is chosen, counting the borders is the same as counting the vertex borders. As a consequence of the previous paragraphs, the following holds: Proposition. In a graph with isoperimetric dimension d$_{iso}$ > 1 , the number of possible borders surrounding p is bounded by where the sum is over the points q which can be enclosed in a border of size b , N$_{q}$ ( b ) is the maximum number of vertex borders of length b , containing p , which can be created starting from the point q , and N$_{Sup}$ ( b ) = sup$_{q}$$_{∈}$$_{V}$$_{(}$$_{b}$$_{)}$ N $^{p}$( b ) . A border is connected if the corresponding vertex border is a connected set. We will need the following proposition regarding N $^{p}$( b ) to obtain a general result. Proposition. The number of connected vertex borders starting from a given point p grows at most exponentially with b : N $^{p}$( b ) ≤ const · C $^{b}$. Proof. A tree is a graph that has no loops. From each connected subgraph A we can draw a number of different spanning trees, i.e. trees having the same set of points P as the original A . For any spanning tree we can construct a path visiting all its vertices in no less than b − 1 steps, and in no more than 2 z$_{Max}$b steps. While the former statement is obvious, we now prove the latter using the following algorithm: starting from i , choose link and cross it; at each vertex on the path, choose a link not yet crossed; if there is no free link, step back 10 µ $^{p}$( b ) ≤ | V ( b ) | ∑ q N$_{q}$ ( b ) ≤ b$^{d}$ iso d frac d iso − 1 · N$_{Sup}$ ( b ) ,