A family of edge-transitive Frobenius metacirculants of small valency

A classification is given for a family of edge-transitive metacirculants of Frobenius groups Z n : Z m of odd order and small valency. In particular, it is shown that, for such a graph Γ , either Aut Γ is soluble and Γ is half-transitive, or Aut Γ = PSL ( 2 , p ) or PGL ( 2 , p ) , and Γ is a known graph of order p q with p and q = p − 1 2 both primes. This leads to new construction of half-transitive graphs.