A characterization of the essential spectrum σess of Schrödinger operators on infinite graphs is derived involving the concept of R-limits. This concept, which was introduced previously for operators on N and Z d as "right-limits", captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate we show that each point in σess(H) corresponds to a bounded generalized eigenfunction of a corresponding R-limit of H. If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds.