normal edge-transitive and $frac{1}{2}-$arc$-$transitive cayley graphs on non-abelian groups of order $2pq$, $p > q$ are odd primes
چکیده
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 $frac{1}{2}-$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 $frac{1}{2}-$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 ≅& langle a, b, c | a^p = b^q = c^2 = e, ac = ca, bc = cb, b^{-1}ab = a^r rangle, or 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 $r^q equiv 1 (mod p)$.