On arc-transitive circulant digraphs
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.
scientific article; zbMATH DE number 1112220