Normal edge-transitive and 12−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 (1) (2013) 213-219.] classified the connected normal edge transitive and‎ ‎12arc-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 ‎12arc-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

Main Subjects


[1] N. Biggs, Algeraic Graph Theory, Cambridge University Press, Cambridge, 1974.

[2] M. R. Darafsheh and A. Assari, Normal edge-transitive Cayley graphs on non-ab elian groups of order 4 p, where p is a prime numb er, Sci. China Math., 56 (2013) 213-219.

[3] X. G. Fang, C. H. Li and M. Y. Xu, On edge-transitive Cayley graphs of valency four, European J. Combin., 25 (2004) 1107-1116.

[4] C. D. Go dsil, On the full automorphism group of a graph, Combinatorica, 1 (1981) 243-256.

[5] P. C. Houlis, Quotients of normal edge-transitive Cayley graphs, MSc Thesis, University of Western Australia, 1998.

[6] C. H. Li, Z. P. Lu and H. Zhang, Tetravalent edge-transitive Cayley graphs with o dd numb er of vertices, J. Combin. Theory Ser. B, 96 (2006) 164-181.

[7] C. E. Praeger, Finite normal edge-transitive Cayley graphs, Bul l. Austral. Math. Soc., 60 (1999) 207-220.

[8] A. A. Talebi, Some normal edge-transitive Cayley graphs on dihedral groups, J. Math. Comput. Sci., 2 (2011) 448-452.

[9] C. Q. Wang, D. J. Wang and M. Y. Xu, On normal Cayley graphs of nite groups, Sci. China Ser. A, 28 (1998) 131-139.

[10] M. Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. , 182 (1998) 309-319.

[11] C. Zhang, J.-X. Zhou and Y.-Q. Feng, Automoprphisms of cubic Cayley graphs of order 2p q, Discrete Math., 309 (2009) 2687-2695.