This article is cited in 5 papers

The action of a group on a graph

V. I. Trofimov


Abstract: 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.

UDC: 512.544.42+519.17

MSC: Primary 05C25, 20B27; Secondary 05C30

Received: 31.01.1984

 English version: Mathematics of the USSR-Izvestiya, 1987, 29:2, 429–447

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