  Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path
algebra of the graph $E$ over a field $K$. The necessary and sufficient con-
ditions are given to assure the existence of a maximal ideal in $L$ and also
the necessary and sufficient conditions on the graph which assure that every
ideal is contained in a maximal ideal is given. It is shown that if a maximal
ideal $M$ of $L$ is non-graded, then the largest graded ideal in $M$ , namely
$gr(M )$, is also maximal among the graded ideals of $L$. Moreover, if $L$ has
a unique maximal ideal $M$ , then $M$ must be a graded ideal. The necessary and
sufficient conditions on the graph for which every maximal ideal is graded, is
discussed.
