  This paper is an attempt to show that, parallel to Elliott's classification
of AF $C^*$-algebras by means of $K$-theory, the graded $K_0$-group classifies
Leavitt path algebras completely. In this direction, we prove this claim at two
extremes, namely, for the class of acyclic graphs (graphs with no cycles) and
comet and polycephaly graphs (graphs which each head is connected to a cycle or
a collection of loops).
