Ising model with spins S=1/2 and 1 on directed and undirected Erdos-Renyi random graphs

Using Monte Carlo simulations we study the Ising model with spin S=1/2 and 1 on {it directed} and {it undirected} Erdos-Renyi (ER) random graphs, with $z$ neighbors for each spin. In the case with spin S=1/2, the {it undirected} and {it directed} ER graphs present a spontaneous magnetization in the universality class of mean field theory, where in both {it directed} and {it 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 {it directed} and {it undirected} ER graphs with spin S=1 we find a first-order phase transition for z=4 and 9 neighbors.