We give necessary and sufficient conditions on a directed graph [Formula: see text] for which the associated Leavit path algebra [Formula: see text] has at least one full idempotent. Also, we define [Formula: see text] sub-graphs of [Formula: see text] and show that [Formula: see text] has at least one full idempotent if and only if there is a sub-graph [Formula: see text] such that the associated Leavitt path algebra [Formula: see text] has at least one full idempotent.