  In this letter we present the inversion of the Mermin-Wagner theorem on
graphs, by proving the existence of spontaneous magnetization at finite
temperature for classical spin models on transient on the average (TOA) graphs,
i.e. graphs where a random walker returns to its starting point with an average
probability $\bar F < 1$. This result, which is here proven for models with
O(n) symmetry, includes as a particular case $n=1$, providing a very general
condition for spontaneous symmetry breaking on inhomogeneous structures even
for the Ising model.
