Williams' conjecture holds for graphs of Gelfand-Kirillov dimension three

A graph of Gelfand-Kirillov dimension three is a connected finite essential graph such that its Leavitt path algebra has Gelfand-Kirillov dimension three. We provide number-theoretic criteria for graphs of Gelfand-Kirillov dimension three to be strong shift equivalent. We then prove that two graphs of Gelfand-Kirillov dimension three 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, $K_0^{\text{gr}}$, are order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphic. As a consequence, we obtain that the Leavitt path algebras of graphs of Gelfand-Kirillov dimension three are graded Morita equivalent if and only if their graph $C^{\ast }$-algebras are equivariant Morita equivalent, and two graphs $E$ and $F$ of Gelfand-Kirillov dimension three are shift equivalent if and only if the singularity categories ${\text{D}}_{\text{sg}}(KE\mathrm{/}{J}_E^2)$ and ${\text{D}}_{\text{sg}}(KF\mathrm{/}{J}_F^2)$ are triangulated equivalent.