Isomorphisms of circulant digraphs
type: Item
instance of: scholarly article
title: Isomorphisms of circulant digraphs
zbMATH Open document ID: 0823.05042
DOI: 10.1007/BF02665228
published in: Applied Mathematics. Series B (English Edition)
publication date: 1995-05-28T00:00:00Z
review text: Let \(\mathbb{Z}_ n\) be the factor group \(\text{mod } n\) of the additive group of integers. The notion of circulant (directed) graph \(\text{DC}_ n(S)\) (on \(n\) vertices) is well known where \(S\) runs through the nonempty subsets of \(\mathbb{Z}_ n- \{0\}\). We write \(S_ 1\simeq S_ 2\) if \(S_ 2\) results from \(S_ 1\) via multiplication by \(r\pmod n\) where \(r\) is relatively prime to \(n\). The reviewer posed the problem whether the isomorphy of \(\text{DC}_ n(S_ 1)\) and \(\text{DC}_ n(S_ 2)\) implies \(S_ 1\simeq S_ 2\) (the converse is evident) [J. Comb. Theory 2, 393 (1967); see also \S\S3-4 in Acta Cybern. 3, 187-214 (1977; Zbl 0374.94037)]. In the present article some properties of minimal generating systems \(S^*\) of \(\mathbb{Z}_ n\) are discussed, the automorphism groups of the graphs \(\text{DC}_ n(S^*)\) are determined, and it is shown that \(\text{DC}_ n(S^*)\) and \(\text{DC}_ n(S)\) are isomorphic only if \(S^*\simeq S\).
Mathematics Subject Classification ID: 05C60
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 749337
zbMATH Keywords: isomorphisms
zbMATH Keywords: circulant digraphs
zbMATH Keywords: isomorphy
zbMATH Keywords: automorphism groups
author: Jixiang Meng
author: Qiongxiang Huang
reviewed by: A. Ádám
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02665228
OpenAlex ID: W165498822
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description: scientific article; zbMATH DE number 749337