A characterization of a family of edge-transitive metacirculant graphs

A characterization is given of the class of edge-transitive Cayley graphs of Frobenius groups Zrd:ZmZrd:Zm with r an odd prime and m   odd, of valency less than 2p12p1 with p1p1 the smallest prime divisor of m  . It is shown that either (rd,m)=(p,p−12)or(29,7), or such a graph is a normal Cayley graph and half-transitive. This provides new construction of half-transitive graphs.