  We introduce a graded homology theory for graded \'etale groupoids. For
$\mathbb Z$-graded groupoids, we establish an exact sequence relating the
graded zeroth-homology to non-graded one. Specialising to the arbitrary graph
groupoids, we prove that the graded zeroth homology group with constant
coefficients $\mathbb Z$ is isomorphic to the graded Grothendieck group of the
associated Leavitt path algebra. To do this, we consider the diagonal algebra
of the Leavitt path algebra of the covering graph of the original graph and
construct the group isomorphism directly. Considering the trivial grading, our
result extends Matui's on zeroth homology of finite graphs with no sinks
(shifts of finite type) to all arbitrary graphs. We use our results to show
that graded zeroth-homology group is a complete invariant for eventual
conjugacy of shift of finite types and could be the unifying invariant for the
analytic and the algebraic graph algebras.
