  Recently it was shown that the notion of flow equivalence of shifts of finite
type in symbolic dynamics is related to the Morita theory and the Grothendieck
group in the theory of Leavitt path algebras \cite{flowa}. In this paper we
show that the notion of conjugacy of shifts of finite type is closely related
to the {\it graded} Morita theory and consequently the {\it graded}
Grothendieck group. This fits into the general framework we have in these two
theories: Conjugacy yields the flow equivalence and the graded Morita
equivalence can be lifted to the Morita equivalence. Starting from a finite
directed graph, the observation that the graded Grothendieck group of the
Leavitt path algebra associated to $E$ coincides with the Krieger dimension
group of the shift of finite type associated to $E$ provides a link between the
theory of Leavitt path algebras and symbolic dynamics. It has been conjectured
that the ordered graded Grothendieck group as $\mathbb Z[x,x^{-1}]$-module (we
call this the graded dimension group) classifies the Leavitt path algebras
completely \cite{hazann}. Via the above correspondence, utilising the results
from symbolic dynamics, we prove that for two purely infinite simple unital
Leavitt path algebras, if their graded dimension groups are isomorphic, then
the algebras are isomorphic.
