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This article is cited in 58 scientific papers (total in 58 papers)
Graphs with polynomial growth
V. I. Trofimov
Abstract:
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.
Received: 19.01.1983
Citation:
V. I. Trofimov, “Graphs with polynomial growth”, Mat. Sb. (N.S.), 123(165):3 (1984), 407–421; Math. USSR-Sb., 51:2 (1985), 405–417
Linking options:
https://www.mathnet.ru/eng/sm2028https://doi.org/10.1070/SM1985v051n02ABEH002866 https://www.mathnet.ru/eng/sm/v165/i3/p407
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Abstract page: | 1112 | Russian version PDF: | 293 | English version PDF: | 23 | References: | 61 |
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