Geodesic
Graphs with polynomial growth
Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 405-417 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\Gamma$ be a connected locally finite vertex-symmetric graph, $R(n)$ the number of vertices of $\Gamma$ at a distance not more than $n$ from some fixed vertex. The equivalence of the following assertions is proved: (a) $R(n)$ is bounded above by a polynomial; (b) there is an imprimitivity system $\sigma$ with finite blocks of $\operatorname{Aut}\Gamma$ on the set of vertices of $\Gamma$ such that $\operatorname{Aut}\Gamma/\sigma$ is finitely generated nilpotent-by-finite and the stabilizer of a vertex of $\Gamma/\sigma$ in $\operatorname{Aut}\Gamma/\sigma$ is finite. Thus, in a certain sense, a description is obtained of the connected locally finite vertex-symmetric graphs with polynomial growth. Bibliography: 8 titles. 
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     author = {V. I. Trofimov},
     title = {Graphs with polynomial growth},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1985_51_2_a5/}
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V. I. Trofimov. Graphs with polynomial growth. Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 405-417. http://geodesic.mathdoc.fr/item/SM_1985_51_2_a5/

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