INTRODUCTION Since the original paper by Ising[1], in which it was proved that the Ising model on an infinite linear chain would not show a phase transition, a huge amount of research has been conducted on the subject. The first phase of this fruitful line investigated regular systems, and after the articles by Peierls[2] and Onsager[3] it became clear that regular lattices in d dimensions would magnetize when d ≥ 2. In a second phase, a large number of fractals was investigated[4–8], mainly via the so- called renormalization group techniques, to discover that, although no rigorous theorem has been proven, those (and only those) fractals which have an infinite minimum order of ramification display spontaneous magnetization. In the same years, fundamental analytical results were obtained for disordered structures embedded in Euclidean lattices, applying percolation theory concepts[9]. More general graphs[10–14] have become increasingly popular in the last twenty years: the main difference with the previous cases is that the metric structure of the embedding space ceases to play an essential role, as in general a graph is a topological structure which is not necessarily embeddable in a finite dimensional Euclidean space. The absence of translational invariance and scale invariance makes general graphs very difficult to study, as ad hoc techniques must be employed, that usually admit no straightforward generalization. An important result would be the identification of a simple parameter, capable of deter- mining whether the Ising model on a given graph exhibits a phase transition: we present here a theorem stating a sufficient condition for a graph to exhibit spontaneous magnetiza- tion, which is a generalization of the classic Peierls-Griffiths theorem[2, 15] for the square lattice. While interesting works, employing the same basic techniques as the Griffiths theo- rem, have been proposed for higher-dimensional lattices, typically stemming from the paper of Dobrushin[16, 17], such as the profound contribution by Isakov[18] and the extensions to non-symmetric situations treated in Pirogov-Sinai theory[19], or directly from the paper by Griffiths, as in Lebowitz and Mazel[20], nothing applying to inhomogeneous networks and arbitrary graphs has yet emerged, and our contribute aims essentially at filling this gap. The reason why the modulus of magnetization 〈| M |〉 is considered is that it is indissolubly tied to the long range order of the graph: it is easy to prove that, when the external field is zero, stating 〈| M |〉 ≥ ϵ > 0 is equivalent to the existence of a non-zero measure subset of 2