On categoricity spectra for locally finite graphs

Under study is the algorithmic complexity of isomorphisms between computable copies of locally finite graphs $G$ (undirected graphs whose every vertex has finite degree). We obtain the following results: If $G$ has only finitely many components then $G$ is $\mathbf{d}$-computably categorical for every Turing degree $\mathbf{d}$ from the class $PA(\mathbf{0}')$. If $G$ has infinitely many components then $G$ is $\mathbf{0}''$-computably categorical. We exhibit a series of examples showing that the obtained bounds are sharp.