normal edge-transitive and $ frac{1}{2}$-arc-transitive cayley graphs on non-abelian groups of order $2pq$ , $p > q$ are primes
چکیده
darafsheh and assari in [normal edge-transitive cayley graphs onnon-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 $frac{1}{2}-$arc-transitive cayley graph of groups of order$4p$. in this paper we continue this work by classifying theconnected cayley graph of groups of order $2pq$, $p > q$ areprimes. as a consequence it is proved that $cay(g,s)$ is a$frac{1}{2}-$edge-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 $t subseteq { cba^{i} | 0 leq i leq p- 1}$ such that $t$ and $t^{-1}$ are orbits of $aut(g,s)$ andbegin{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 $r^q equiv 1 (mod p)$.