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Cubic symmetric graphs of order twice an odd prime-power

Part of: Graph theory Permutation groups

Published online by Cambridge University Press:  09 April 2009

Yan-Quan Feng  and
Jin Ho Kwak
Yan-Quan Feng
Affiliation:
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R., China, e-mail: yqfeng@center.njtu.edu.cn
Jin Ho Kwak
Affiliation:
Combinatorial and Computational, Mathematics Center, Pohang University of Science and Technology, Pohang, 790-784, Korea, e-mail: jinkwak@postech.ac.kr
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Abstract

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An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.

Keywords

Symmetric graphss-regular graphss-arc-transitive graphs.

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 2 , October 2006 , pp. 153 - 164
Copyright
Copyright © Australian Mathematical Society 2006

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