Generalization of the Peierls-Griffiths Theorem for the Ising Model on Graphs Riccardo Campari ∗ and Davide Cassi † Dipartimento di Fisica, Universit` a di Parma, Viale Usberti 7/A, 43100 Parma, Italy (Dated: November 11, 2018) We present a sufficient condition for the presence of spontaneous magnetization for the Ising model on a general graph, related to its long-range topology. Applying this condition we are able to prove the existence of a phase transition at temperature T > 0 on a wide class of general networks. The possibility of further extensions of our results is discussed. Abstract arXiv:1002.1227v1 [cond-mat.stat-mech] 5 Feb 2010 ∗ riccardo.campari@fis.unipr.it † davide.cassi@fis.unipr.it 1