On Solvable Groups and Circulant Graphs

Let ϕ be Euler’s phi function. We prove that a vertex-transitive graphΓ of order n, with gcd(n, ϕ(n)) = 1, is isomorphic to a circulant graph of order n if and only if Aut(Γ) contains a transitive solvable subgroup. As a corollary, we prove that every vertex-transitive graph Γ of order n is isomorphic to a circulant graph of order n if and only if for every such Γ,Aut (Γ) contains a transitive solvable subgroup and n = 4, 6, or gcd(n, ϕ(n)) = 1.