Ising model on the scale-free network with a Cayley-tree-like structure

We derive an exact expression for the magnetization and the zero-field susceptibility of the Ising model on a random graph with degree distribution P(k)proportional to k(-gamma) and with a boundary consisting of leaves, that is, vertices whose degree is 1. The system has no magnetization at any finite temperature, and the susceptibility diverges below a certain temperature T-s depending on the exponent gamma. In particular, T-s reaches infinity for gamma <= 4. These results are completely different from those of the case having no boundary, indicating the nontrivial roles of the leaves in the networks.