Williams' Conjecture holds for meteor graphs

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 $C^{\ast }$-algebras are equivariant Morita equivalent.