1-regular Cayley graphs of valency 7 (Q2872016)

We say that a graph \(\Gamma\) is \(1\)-regular if its automorphism group \(\mathrm{Aut}(\Gamma)\) acts regularly on its arcs. The authors present the classification of the \(1\)-regular Cayley graphs of valency \(7\). Their main result (Theorem \(1.1\)) states that every such graph is connected and satisfies one of the following (\(N=\mathrm{Core}_A(G)\)): (1) \(G=N\) and \(\Gamma\) is a normal Cayley graph, or (2) \(|G:N|=2\) and \(\Gamma\) is a bi-normal Cayley graph, or (3) \(\Gamma\) is a normal cover of a core-free graph (up to isomorphism): \((A,G)=(S_7,S_6)\).