Ising model with spins S=1/2 and 1 on directed and undirected Erdös–Rényi random graphs

Using Monte Carlo simulations, we study the Ising model with spin S=1/2 and 1 on directed and undirected Erdös–Rényi (ER) random graphs, with z neighbors for each spin. In the case with spin S=1/2, the undirected and directed ER graphs present a spontaneous magnetization in the universality class of mean field theory, where in both directed and undirected ER graphs the model presents a spontaneous magnetization at p=z/N(z=2,3,…,N), but no spontaneous magnetization at p=1/N which is the percolation threshold. For both directed and undirected ER graphs with spin S=1, we find a first-order phase transition for z=4 and 9 neighbors.