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 ℤd. In this article we prove that for each d > 1 the set of Cayley graphs of ℤd 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 ℤd that are limits of minimal vertex-primitive graphs. The proofs rely on a connection of the automorphism groups of Cayley graphs of ℤd with crystallographic groups.
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Original Russian Text Copyright © 2007 Kostousov K. V.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 3, pp. 606–620, May–June, 2007.
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Kostousov, K.V. The Cayley graphs of ℤd and the limits of vertex-primitive graphs of HA-type. Sib Math J 48, 489–499 (2007). https://doi.org/10.1007/s11202-007-0051-z
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DOI: https://doi.org/10.1007/s11202-007-0051-z