  Suppose that $R$ is an associative unital ring and that $E=(E^0,E^1,r,s)$ is
a directed graph. Utilizing results from graded ring theory we show, that the
associated Leavitt path algebra $L_R(E)$ is simple if and only if $R$ is
simple, $E^0$ has no nontrivial hereditary and saturated subset, and every
cycle in $E$ has an exit. We also give a complete description of the center of
a simple Leavitt path algebra.
