Finite vertex-primitive edge-transitive metacirculants.
type: Item
instance of: scholarly article
title: Finite vertex-primitive edge-transitive metacirculants.
zbMATH Open document ID: 1319.20002
author: Cai Heng Li
author: Zai Ping Lu
author: Jiangmin Pan
published in: Journal of Algebraic Combinatorics
publication date: 2014-11-19T00:00:00Z
review text: Let \(\Gamma=(V,E)\) be a connected, undirected, simple graph. Then \(\Gamma\) is called a metacirculant if its automorphism group \(\Aut(\Gamma)\) contains a metacyclic group \(R\) which is transitive on the vertex set \(V\) (that is, \(R\) is a cyclic extension of a cyclic normal subgroup). It is a long standing problem to characterize metacirculants for which \(\Aut(\Gamma)\) is transitive on the edge set \(E\). The main result of the present paper is to solve this problem in the case that \(\Aut(\Gamma)\) acts primitively on \(V\). More precisely, assume that \(G\) is a subgroup of \(\Aut(\Gamma)\) such that \(R\leq G\) and that \(G\) is both transitive on \(E\) and primitive on \(V\), then there are nine classes of metacirculants which can arise. In particular, \(\Gamma\) is a Cayley graph except in the following cases: (i) \(\Gamma\) is the line graph of the complete graph \(K_p\) (or the complement of this line graph) for some prime \(p\equiv 1\pmod 4\); (ii) \(R=\mathbb Z_{17}:\mathbb Z_8\) and \(\Aut(\Gamma)=\mathrm{P}\Gamma\)L
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 6371198
zbMATH Keywords: metacirculants
zbMATH Keywords: edge transitive graphs
zbMATH Keywords: primitive permutation groups
MaRDI profile type: Publication
full work available at URL: s10801-014-0507-8
OpenAlex ID: W2076403724
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DOI: 10.1007/S10801-014-0507-8
description: scientific article; zbMATH DE number 6371198