Geodesic
The action of a~group on a~graph
Izvestiya. Mathematics , Tome 29 (1987) no. 2, pp. 429-447.

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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. 
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V. I. Trofimov. The action of a~group on a~graph. Izvestiya. Mathematics , Tome 29 (1987) no. 2, pp. 429-447. http://geodesic.mathdoc.fr/item/IM2_1987_29_2_a7/

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