1992
DOI: 10.1007/bf00053384
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On the action of a group on a graph

Abstract: The present paper was written at the request of one of the editors for a survey on some results of the author. The research presents an approwch to studying groups acting on connected vertex-symmetric graphs of finite valency. This situation arises naturally when studying groups (the action of a group on its Cayley graph, the action of a primitive group on the corresponding permutation graphs, etc.) and also when studying certain applications (description of graphs with given symmetry properties and some other… Show more

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Cited by 8 publications
(5 citation statements)
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“…The first of the three following results is [4], Theorem 1; the second and third are [9], Corollary 1 and Proposition 2.3.…”
Section: Preliminariesmentioning
confidence: 83%
See 1 more Smart Citation
“…The first of the three following results is [4], Theorem 1; the second and third are [9], Corollary 1 and Proposition 2.3.…”
Section: Preliminariesmentioning
confidence: 83%
“…The following results, obtained by Trofimov [4,9], play a crucial role in all investigations of graphs with polynomial growth. To formulate them we need an additional definition:…”
Section: Preliminariesmentioning
confidence: 91%
“…It is easy to see that Theorem 1 is no longer true for groups with infinite vertex stabilizers. In [3], the case of a connected locally finite graph admitting a vertex-transitive group of o-automorphisms (with finite or infinite vertex stabilizers) was investigated. The proof of Theorem 1 is similar to the proof of this result in [3].…”
Section: Final Commentsmentioning
confidence: 99%
“…For any nonnegative integer n, denote by ξ g n the maximum of d y g y as y runs the set of vertices at a distance of at most n from x in . Obviously, ξ g n ≤ 2n + C where C = d x g x is independent of n. In the case ξ g n = o n (i.e., ξ g n /n → 0 as n → ∞; this property of g is independent of the choice of x), g is called an o-automorphism of (see [3]). For G ≤ Aut , o G denotes the normal subgroup of G consisting of all o-automorphisms of contained in G. Thus o Aut is the set of all o-automorphisms of , and o G = G ∩ o Aut for G ≤ Aut .…”
Section: Introductionmentioning
confidence: 98%
“…[14,16,6,15,12,11,2,1]). This motivated us to study the general question: How many geodetic rays does a graph contain?…”
Section: Introductionmentioning
confidence: 98%