On solvable groups and circulant graphs
type: Item
instance of: scholarly article
title: On solvable groups and circulant graphs
zbMATH Open document ID: 0973.05039
author: Edward Dobson
published in: European Journal of Combinatorics
publication date: 2001-12-03T00:00:00Z
review text: Solvable graphs are defined to be graphs whose automorphism group contains a solvable subgroup. A circulant graph of order \(n\) has an automorphism group which contains an \(n\)-cycle. In this paper every vertex-transitive graph \(\Gamma\) of order \(n\) with \(\text{gcd}(n,\varphi(n))= 1\) is proved to be isomorphic to a circulant graph of order \(n\) if and only if \(\Gamma\) is a solvable graph. This result generalizes an analogous theorem of Marušič which is restricted to the case that \(n= pq\) is the product of two distinct prime numbers. As a corollary, every vertex-transitive graph of order \(n\) is stated to be isomorphic to a Cayley graph of order \(n\) if and only if every vertex-transitive graph of order \(n\) is solvable.
reviewed by: Ulrike Baumann
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 1538570
zbMATH Keywords: automorphism group
zbMATH Keywords: circulant graph
zbMATH Keywords: vertex-transitive graph
zbMATH Keywords: solvable graph
zbMATH Keywords: Cayley graph
MaRDI profile type: Publication
OpenAlex ID: W1982321442
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DOI: 10.1006/EUJC.2000.0412
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description: scientific article; zbMATH DE number 1538570