Normal edge-transitive and 12 12-arc-transitive Cayley graphs on non-abelian groups of order 2pq 2pq, p>q p>q are primes

Darafsheh and Assari in [Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number, Sci. China Math. {\bf 56} (1) (2013) 213− − 219.] classified the connected normal edge transitive and frac12− frac12− arc-transitive Cayley graph of groups of order 4p 4p . In this paper we continue this work by classifying the connected Cayley graph of groups of order 2pq 2pq , p>q p>q are primes. As a consequence it is proved that Cay(G,S) Cay(G,S) is a frac12− frac12− edge-transitive Cayley graph of order 2pq 2pq , p>q p>q if and only if |S| |S| is an even integer greater than 2, S=TcupT −1 S=TcupT−1 and Tsubseteqcba i |0leqileqp−1 Tsubseteqcbai|0leqileqp−1 such that T T and T −1 T−1 are orbits of Aut(G,S) Aut(G,S) and begin{eqnarray*} G &=& langle a, b, c | a^p = b^q = c^2 = e, ac = ca, bc = cb, b^{-1}ab = a^r rangle, G &=& langle a, b, c | a^p = b^q = c^2 = e, c ac = a^{-1}, bc = cb, b^{-1}ab = a^r rangle, end{eqnarray*} where r q equiv1(modp) rqequiv1(modp).