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ON GRAPHS OF PRIME VALENCY ADMITTING A SOLVABLE ARC-TRANSITIVE GROUP

Part of: Permutation groups Algebraic combinatorics

Published online by Cambridge University Press:  13 May 2015

BOŠTJAN KUZMAN
BOŠTJAN KUZMAN*
Affiliation:
University of Ljubljana, Faculty of Education, Department of Math and Computer Science, Kardeljeva ploščad 16, 1000 Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia email [email protected]
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Abstract

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Let $X$ be a simple, connected, $p$-valent, $G$-arc-transitive graph, where the subgroup $G\leq \text{Aut}(X)$ is solvable and $p\geq 3$ is a prime. We prove that $X$ is a regular cover over one of the three possible types of graphs with semi-edges. This enables short proofs of the facts that $G$ is at most 3-arc-transitive on $X$ and that its edge kernel is trivial. For pentavalent graphs, two further applications are given: all $G$-basic pentavalent graphs admitting a solvable arc-transitive group are constructed and an example of a non-Cayley graph of this kind is presented.

Keywords

arc-transitive graphssolvable groupedge kernelpentavalent graphs

MSC classification

Primary: 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Secondary: 05E18: Group actions on combinatorial structures
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 2 , October 2015 , pp. 214 - 227
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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