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ON GRAPHS OF PRIME VALENCY ADMITTING A SOLVABLE ARC-TRANSITIVE GROUP
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 92 / Issue 2 / October 2015
- Published online by Cambridge University Press:
- 13 May 2015, pp. 214-227
- Print publication:
- October 2015
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- Article
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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.
