  A meteor graph is a connected graph with no sources and sinks consisting of
two disjoint cycles and the paths connecting these cycles. We prove that two
meteor graphs are shift equivalent if and only if they are strongly shift
equivalent, if and only if their corresponding Leavitt path algebras are graded
Morita equivalent, if and only if their graded $K$-theories, $K_0^{gr}$, are
$\mathbb Z [x,x^{-1}]$-module isomorphic. As a consequence, the Leavitt path
algebras of meteor graphs are graded Morita equivalent if and only if their
graph $C^*$-algebras are equivariant Morita equivalent.
