On solvable groups and circulant graphs (Q1587909)

Solvable graphs are defined to be graphs whose automorphism group contains a solvable subgroup. A circulant graph of order \(n\) has an automorphism group which contains an \(n\)-cycle. In this paper every vertex-transitive graph \(\Gamma\) of order \(n\) with \(\text{gcd}(n,\varphi(n))= 1\) is proved to be isomorphic to a circulant graph of order \(n\) if and only if \(\Gamma\) is a solvable graph. This result generalizes an analogous theorem of Marušič which is restricted to the case that \(n= pq\) is the product of two distinct prime numbers. As a corollary, every vertex-transitive graph of order \(n\) is stated to be isomorphic to a Cayley graph of order \(n\) if and only if every vertex-transitive graph of order \(n\) is solvable.