Using rationality, like in language theory, we define a family of infinite graphs. This family is a strict extension of the context-free graphs of Muller and Schupp, the equational graphs of Courcelle and the prefix recogniza- ble graphs of Caucal. We give basic properties, as well as an internal and an external characterization of these graphs. We also show that their traces form an AFL of recursive languages, containing the context-free languages.