A classification of 2-arc-transitive circulants
A graph \(X\) is said to be 2-arc-transitive if \(\Aut(X)\) acts transitively on the arcs of \(X\). Let \(n\) be a positive integer and \(S\subseteq Z_n\backslash \{0\}\) where \(i\in S\) iff \(n- i\in S\). The graph \(X(n; S)\) is called a circulant if the vertices are the elements of \(Z_n\) and there is an edge joining \(i\) and \(j\) iff \(i- j\in S\). The following classification of 2-arc-transitive graphs is obtained: A connected, 2-arc-transitive circulant of order \(n\), \(n\geq 3\), is one of the following graphs: (i) the complete graph \(K_n\), which is exactly 2-transitive; (ii) the complete bipartite graph \(K_{n/2, n/2}\), \(n\geq 6\), which is exactly 3-transitive; (iii) the complete bipartite graph \(K_{n/2, n/2}\) minus a 1-factor, \({n\over 2}\geq 5\) odd, which is exactly 2-transitive; and (iv) the cycle \(C_n\) of length \(n\), which is \(k\)-arc-transitive for all \(k\geq 0\).
scientific article; zbMATH DE number 887234