The talk is concerned with semilinear elliptic inequalities with a potential on infinite weighted graphs. Given a distance on the graph, we assume an upper bound on its Laplacian, and a growth condition on a suitable weighted volume of balls. Under such hypotheses, we discuss the nonexistence of nonnegative nontrivial solutions. We also see the optimality of our conditions. Such results have been recently obtained jointly with D.D. Monticelli and J. Somaglia.