for which the number of boundaries µ $^{p}$( b ) is greater than any exponential (e.g. a semi-infinite line connected to a point on a plane), the hypothesis of the theorem is false but the graph as a whole can still magnetize. 

An important, yet straightforward, observation is that whenever a subgraph of non zero measure exists that is magnetizable, all the graph is magnetizable: in fact all the correlation functions, as computed on the subgraph, are smaller than or equal to the corresponding ones in the complete graph; when, on the other hand, the graph is formed by a collection of zero measure, weakly connected, magnetizable subgraphs (e.g. an infinite collection of parallel planes, each connected via a single link to the next one), there is no guarantee that 〈| M |〉 > 0. 

Our result about the Ising model on graphs is a further step towards a full comprehension of the mechanism of phase transitions on general networks: together with a sufficient condi- tion for the lack of spontaneous magnetization[22], it allows to ascertain the magnetizability of a large number of structures with a minimal amount of computation. 

Further steps extending this work should aim at closing the gap between magnetizable and non-magnetizable graphs under the 〈| M |〉 definition, in order to identify a condition both necessary and sufficient for spontaneous magnetization; another direction of development could be to treat non symmetric situations, as in Pirogov-Sinai theory. 

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