We study patterns of the phase diagram of ferromagnetic Ising models on graphs under an external magnetic field. We provide an example of a tree with only two types of vertices on which for a range of values of the external field there is a unique Gibbs distribution at low enough and at high enough temperatures, while at intermediate temperatures there is phase coexistence (in other words, a reentrance transition takes place).
"Lack of monotonicity in ferromagnetic Ising model phase diagrams." Ann. Appl. Probab. 8 (1) 234 - 245, February 1998. https://doi.org/10.1214/aoap/1027961042