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 Izv. Akad. Nauk SSSR Ser. Mat., 1986, Volume 50, Issue 5, Pages 1077–1096 (Mi izv1564)

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.

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

Bibliographic databases:

UDC: 512.544.42+519.17
MSC: Primary 05C25, 20B27; Secondary 05C30

Citation: V. I. Trofimov, “The action of a group on a graph”, Izv. Akad. Nauk SSSR Ser. Mat., 50:5 (1986), 1077–1096; Math. USSR-Izv., 29:2 (1987), 429–447

Citation in format AMSBIB
\Bibitem{Tro86} \by V.~I.~Trofimov \paper The action of a~group on a~graph \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1986 \vol 50 \issue 5 \pages 1077--1096 \mathnet{http://mi.mathnet.ru/izv1564} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=873661} \zmath{https://zbmath.org/?q=an:0676.05044|0616.05040} \transl \jour Math. USSR-Izv. \yr 1987 \vol 29 \issue 2 \pages 429--447 \crossref{https://doi.org/10.1070/IM1987v029n02ABEH000977} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. I. Trofimov, “Asymptotic behavior of automorphisms of graphs”, Math. USSR-Sb., 62:1 (1989), 277–287
2. R MOLLER, “Topological groups, automorphisms of infinite graphs and a theorem of Trofimov”, Discrete Mathematics, 178:1-3 (1998), 271
3. Vladimir I. Trofimov, “Kernels of van den Dries–Wilkie Homomorphisms and o-Automorphisms of Graphs”, Journal of Algebra, 226:2 (2000), 967
4. V.I. Trofimov, “On the action of a group on a graph, II”, Discrete Mathematics, 2010
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