ملخص المقالة :
For two normal edge-transitive Cayley graphs on groups H and K which have nocommon direct factor and $\gcd(|H/H^\prime|,|Z(K)|)=1=\gcd(|K/K^\prime|,|Z(H)|)$, we consider fourstandard products of them and it is proved that only tensor product of factors can be normaledge-transitive.
المصادر:
[1] M. Alaeiyan. On normal edge-transitive Cayley graphs of some abelian groups. Southeast Asian Bull. Math. 33 (2009), no. 1, 13-19.
[2] M. R. Darafsheh, A. Assari. Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number. Sci. China Math. 56 (2013), no. 1, 213-219.
[3] J. N. S. Bidwell, M. J. Curran, D. J. McCaughan. Automorphisms of direct products of nite groups. Arch. Math. (Basel) 86 (2006), no. 6, 481-489.
[4] G. B. Cagaanan, S. R. J. Canoy. On the hull sets and hull number of the Cartesian product of graphs. Discrete Math. 287 (2004), no. 1-3, 141-144.
[5] P. Dorbec, M. Mollard, S. Klavzar, S. Spacapan. Power domination in product graphs. SIAM J. Discrete Math. 22 (2008), no. 2, 554-567.
[6] X. G. Fang, C. H. Li, M. Y. Xu. On edge-transitive Cayley graphs of valency four. European J. Combin. 25 (2004), no. 7, 1107-1116.
[7] C. D. Godsil. On the full automorphism group of a graph. Combinatorica 1 (1981), no. 3, 243-256.
[8] C. Godsil, G. Royle. Algebraic graph theory. Graduate Texts in Mathematics, 207. Springer-Verlag, New York, 2001.
[9] P. C. Houlis. Quotients of normal edge-transitive Cayley graphs. University of Western Australia, 1998.
[10] N. Hosseinzadeh, A. Assari. Graph operations on Cayley graphs of semigroups. International Journal of Applied Mathematical Research, 3 (1) (2014) 54-57.
[11] C. H. Li, Z. P. Lu, H. Zhang. Tetravalent edge-transitive Cayley graphs with odd number of vertices. J. Combin. Theory Ser. B 96 (2006), no. 1, 164-181.
[12] D. Marusic, R. Nedela. Maps and half-transitive graphs of valency 4. European J. Combin. 19 (1998), no. 3, 345-354.
[13] C. E. Praeger. Finite normal edge-transitive Cayley graphs. Bull. Austral. Math. Soc. 60 (1999), no. 2, 207-220.
[14] C. Wang, D. Wang, M. Xu. Normal Cayley graphs of nite groups. Sci. China Ser. A 41 (1998), no. 3, 242-251.
[15] M. Y. Xu. Automorphism groups and isomorphisms of Cayley digraphs. Graph theory (Lake Bled, 1995). Discrete Math. 182 (1998), no. 1-3, 309-319.
[16] J. M. Xu, C. Yang. Connectivity and super-connectivity of Cartesian product graphs. Ars Combin. 95 (2010), 235-245.
[17] J. M. Xu, C. Yang. Connectivity of Cartesian product graphs. Discrete Math. 306 (2006), no. 1, 159-165.