A family of edge-transitive Frobenius metacirculants of small valency

A classification is given for a family of edge-transitive metacirculants of Frobenius groups Zn:Zm of odd order and small valency. In particular, it is shown that, for such a graph Γ, either AutΓ is soluble and Γ is half-transitive, or AutΓ=PSL(2,p) or PGL(2,p), and Γ is a known graph of order pq with p and q=p−12 both primes. This leads to new construction of half-transitive graphs.