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Sibirskii Matematicheskii Zhurnal, 2007, Volume 48, Number 3, Pages 606–620
(Mi smj51)
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This article is cited in 7 scientific papers (total in 7 papers)
The Cayley graphs of $\mathbb Z^d$ and the limits of vertex-primitive graphs of $HA$-type
K. V. Kostousov Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Abstract:
We study the limits of the finite graphs that admit some vertex-primitive group of automorphisms with a regular abelian normal subgroup. It was shown in [1] that these limits are Cayley graphs of the groups $\mathbb Z^d$. In this article we prove that for each $d>1$ the set of Cayley graphs of $\mathbb Z^d$ presenting the limits of finite graphs with vertex-primitive and edge-transitive groups of automorphisms is countable (in fact, we explicitly give countable subsets of these limit graphs). In addition, for $d<4$ we list all Cayley graphs of $\mathbb Z^d$ that are limits of minimal vertex-primitive graphs. The proofs rely on a connection of the automorphism groups of Cayley graphs of $\mathbb Z^d$ with crystallographic groups.
Keywords:
vertex-primitive graph, edge-transitive graph, limit graph, Cayley graph of a finite rank free abelian group, crystallographic group.
Received: 21.01.2006
Citation:
K. V. Kostousov, “The Cayley graphs of $\mathbb Z^d$ and the limits of vertex-primitive graphs of $HA$-type”, Sibirsk. Mat. Zh., 48:3 (2007), 606–620; Siberian Math. J., 48:3 (2007), 489–499
Linking options:
https://www.mathnet.ru/eng/smj51 https://www.mathnet.ru/eng/smj/v48/i3/p606
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Abstract page: | 428 | Full-text PDF : | 208 | References: | 50 |
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