# Normal Edge-Transitive and $\frac{1}{2}-$Arc$-$Transitive Cayley Graphs on Non-Abelian Groups of Order $2pq$, $p > q$ are Odd Primes

Document Type : Research Paper

Authors

University of Kashan

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 (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

$G ≅ < a, b, c | a^p = b^q = c^2 = e, ac = ca, bc = cb, b^{-1}ab = a^r >$,  or

$G ≅ < a, b, c | a^p = b^q = c^2 = e, c ac = a^{-1}, bc = cb, b^{-1}ab = a^r >$,

where $r^q \equiv 1 (mod p)$.

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