Geodesic
Automorphisms of graphs and a~characterization of lattices
Izvestiya. Mathematics , Tome 22 (1984) no. 2, pp. 379-391.

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An automorphism $g$ of an undirected connected graph $\Gamma$ is called bounded if for some natural number $c$and an arbitrary vertex $\alpha$ of the graph $\Gamma$ the inequality $d(\alpha,g(\alpha))$. The structure of vertex-transitive groups of bounded automorphisms of locally finite graphs is studied. A characterization of locally finite graphs which admit a vertex-transitive group of bounded automorphisms is obtained. Bibliography: 2 titles. 
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V. I. Trofimov. Automorphisms of graphs and a~characterization of lattices. Izvestiya. Mathematics , Tome 22 (1984) no. 2, pp. 379-391. http://geodesic.mathdoc.fr/item/IM2_1984_22_2_a10/

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[2] Neumann B. H., “Groups with finite classes of conjugate elements”, Proc. London Math. Soc., 1 (1951), 178–187 | DOI | MR | Zbl