Isomorphisms of circulant digraphs of degree 3
The circulant directed graph with \(n\) vertices and with characteristic set \(S = \{s_1, s_2, \dots, s_k\}\) is denoted by \(C_n (S)\). An isomorphism \(f\) from \(C_n (S)\) to \(C_n (T)\) (where \(|S |= |T |)\) is called zero-preserving if \(f(0) = 0\); among these, \(f\) is said to be special if \((\forall i)\) \([f(i) \equiv \lambda i \pmod n]\) for a \(\lambda\) which is relatively prime to \(n\). In Theorem 1 the graph pairs \((C_n (S), C_n (T))\) are characterized for which any zero-preserving isomorphism from \(C_n (S)\) to \(C_n (T)\) is a special isomorphism. The main results of the paper (Theorems 2 and 3) concern the particular case \(k (= |S |) = 3\). They are stated after a sequence of lemmas. The (three-element) sets \(S\) for which there exists a zero-preserving non-special isomorphism from \(C_n (S)\) are completely determined. Some consequences of Theorem 1 for the automorphisms of \(C_n (S)\) are also derived.
scientific article; zbMATH DE number 913300