IUST International Journal of Engineering Science, Vol. 19, No.1-2, 2008, Page 1-11 Mathematics & Industrial Engineering Special Issue NORMAL 6-VALENT CAYLEY GRAPHS OF ABELIAN GROUPS Mehdi Alaeiyan Abstract: We call a Cayley graph Γ = Cay (G, S) normal for G, if the right regular representation R(G) of G is normal in the full automorphism group of Aut(Γ). In this paper, a classification of all non-normal Cayley graphs of finite abelian group with valency 6 was presented. Keywords: Cayley graph, normal Cayley graph, automorphism group. 1. Introduction 1 Let G be a finite group, and S be a subset of G not containing the identity element 1 G . The Cayley digraph Γ=Cay(G,S) of G relative to S is defined as the graph with vertex set V(Γ) = G and edge set E(Γ) consisting of those ordered pairs (x, y) from G for which yx -1 ∈ S. Immediately from the definition we find that, there are three obvious facts: (1) Aut(Γ) contains the right regular representation R(G) of G and so Γ is vertex- transitive. (2) Γ is connected if and only if G =< S>. (3) Γ is an undirected if and only if S -1 = S. A Cayley (di)graph Γ=Cay(G,S) is called normal if the right regular representation R(G) of G is a normal subgroup of the automorphism group of Γ. The concept of normality of Cayley (di)graphs is known to be important for the study of arc-transitive graphs and half-transitive graphs (see[1,2]). Given a finite group G, a natural problem is to determine all normal or non-normal Cayley (di)graphs of G. This problem is very difficult and is solved only for the cyclic groups of prime order by Alspach [3] and the groups of order twice a prime by Du et al. [4], while some partial answers for other groups to this problem can be found in [5-8]. Wang et al. [8] characterized all normal disconnected Cayley’s graphs of finite groups. Therefore the main work to determine the normality of Cayley graphs is to determine the normality of connected Cayley graphs. In [5, 6], all non-normal Cayley graphs of abelian groups with valency at most 5 were classified. The purpose of this paper is the following main theorem. Theorem 1.1 Let Γ = Cay (G, S) be a connected undirected Cayley graph of a finite abelian group G on S with valency 6. Then Γ is normal except when one of the following cases happens: Paper first received March. 2, 2007, and in revised form Jan.10, 2007. M Alaeiyan, Department of Mathematics, Iran University of Science and Technology, alaeiyan@iust.ac.ir (1): G = = <a> × <b> × <c> × <d> ×<e>, 5 2 z S = {a, b, c, abc, d, e}. (2): G = × Z m = <a> × <b> × <c> × <d> ( 3), 3 2 z m ≥ S = {a, b, c, abc, d, d -1 }. (3): G = × Z 4 = <a> × <b> × <c>, 2 2 z S = {a, b, ab, c 2 , c, c -1 }. (4): G = × Z 4 = <a> × <b> × <c> × <d> × <e>, 4 2 z S = {a, b, c, d, e, e -1 }. (5): G = × Z 4 = <a> × <b> × <c> × <d> 3 2 z S 1 = {a, b, c, d 2 , d, d -1 }, S 2 = {a, b, ab, c, d, d -1 }, S 3 = {a, b, c, ad 2 , d, d -1 }. (6): G = × Z 6 = <a> × <b> × <c>, 2 2 z S = {a, b, ab, c 3 , c, c -1 }. (7): G = × Z 6 = <a> × <b> × <c> × <d>, 3 2 z S = {a, b, c, d 3 , d, d -1 }. (8): G = Z 6 × Z 2m = <a> × <b> ( m 2), ≥ S = {a 3 , b m , a, a -1 , b, b -1 }. (9): G = Z 2 × Z 6 × Z m = <a> × <b> × <c> ( 3), m ≥ S = {a, b 3 , b, b -1 , c, c -1 }. (10): G = Z 4 × Z 2m = <a> × <b> ( 2), m ≥ S = {a, a -1 , a 2 , b, b -1 , b m }. (11): G = Z 2 × Z 4 × Z m = <a> × <b> × <c> ( 3), m ≥ S 1 = {a, b, b -1 , b 2 , c, c -1 }, S 2 = {a, b, b -1 , ab 2 , c, c -1 }. (12): G = Z 2 × Z 4 × Z 2m = <a> × <b> × <c> ( 2), m ≥ S = {a, b, b -1 , c, c -1 , c m }. (13): G = × Z 4 × Z m = <a> × <b> × <c> × <d> (m≥3), S = {a, b, c, c -1 , d, d −1 }. 2 2 z