Abstract :
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., 56 (1) (2013) 213-219.] classified the connected normal edge transitive and 12−arc-transitive Cayley graph of groups of order 4p. In this paper we continue this work by classifying the connected Cayley graph of groups of order 2pq, p q are primes. As a consequence it is proved that Cay(G,S) is a 12−arc-transitive Cayley graph of order 2pq, p q if and only if |S| is an even integer greater than 2, S = T \cup T^{-1} and T \subseteq \{ cb^ja^{i} \ | \ 0 \leq i \leq p - 1\}, 1 \leq j \leq q-1, such that T and T^{-1} are orbits of Aut(G,S) and \begin{eqnarray*} G \cong \langle a, b, c \ | \ a^p = b^q = c^2 = e, ac = ca, bc = cb, b^{-1}ab = a^r \rangle, \ or\\ G \cong \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 \equiv 1 \ (mod p).
Keywords :
Cayley graph , normal edge , transitive , normal arc , transitive
