On arc-transitive circulant digraphs
type: Item
instance of: scholarly article
title: On arc-transitive circulant digraphs
zbMATH Open document ID: 0891.05037
published in: Applied Mathematics. Series B (English Edition)
publication date: 1998-07-08T00:00:00Z
review text: Let \(Z_n\) be the cyclic group of order \(n\). For \(S\subseteq Z_n \setminus \{0\}\), a circulant digraph \(C_n(S)\) is a directed simple graph with vertex set \(Z_n\) and arc set \(\{(u,v) \mid v-u\in S\}\). The following results are presented: Theorem 1. Let \(S= \{a,b\}\) generate \(Z_n\). Then \(C_n(S)\) is arc-transitive iff \(S=S+ (a-b)\) or \(S=aS^*\) \((S^*\leq Z^*_n)\). Theorem 2. Let \(C_n(S)\) \((S\neq -S)\) be a connected circulant digraph with \(X_0|S\) being the symmetric group on \(S\). Then the set \(S\) is determined by one of the following: (i) \(S=B+a\), where \(\varnothing \neq B<Z_n\) and \(a\in Z_n\). (ii) \(S=aS^*\), where \(S^*\) is a two-element subgroup of \(Z^*_n\). (iii) \(S=B+u^* \setminus \{u^*\}\), where \(B<Z_n\) with \(|B|\geq 4\) and \(u^* \in Z_n \setminus B\) with \(o(u^*)=n/ |B|\). Moreover, all the arc-transitive digraphs of degree 2 and 3 are given.
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 05C20
zbMATH DE Number: 1112220
zbMATH Keywords: circulant digraph
zbMATH Keywords: arc-transitive digraphs
author: Qiongxiang Huang
reviewed by: Ulrike Baumann
MaRDI profile type: Publication
full work available at URL: s11766-997-0049-0
OpenAlex ID: W2317005925
cites work: Isomorphism of circulant graphs and digraphs
cites work: Circulants and their connectivities
cites work: Reliable circulant networks with minimum transmission delay
cites work: On the Classification of Symmetric Graphs with a Prime Number of Vertices
cites work: On the isomorphisms and automorphism groups of circulants
cites work: Q5512231
DOI: 10.1007/S11766-997-0049-0
description: scientific article; zbMATH DE number 1112220