The action of a~group on a~graph

V. I. Trofimov

Izvestiya. Mathematics , Tome 29 (1987) no. 2, pp. 429-447

Résumé

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.