The action of a group on a graph

A classification of automorphisms of a connected graph $\Gamma$ is given. In particular, an automorphism $g$ is called an $o$-automorphism if for some (and then also for any) vertex $x$ of the graph $\Gamma$
$$ \max\{d_\Gamma(y,g(y))\mid y\in V(\Gamma),\ d_\Gamma(x,y)\leqslant n\}=o(n). $$

It is proved that a connected locally finite graph admits a vertex-transitive group of $o$-automorphisms if and only if the graph is a nilpotent lattice.
Bibliography: 9 titles.