Abstract
For a group G, and a subset S of G such that 1 G ∉ S, let X = C a y(G,S) be the corresponding Cayley graph. Then X is said to be normal edge transitive if NA u t(X)(G) is transitive on edges. In this paper, we determine all connected directed Cayley graphs of finite abelian groups with valency at most 3 which are normal edge transitive but not normal.
AMS
05C10; 05C25
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Introduction
Throughout this paper, graphs are finite, simple and directed. For a graph X, let V(X), E(X) and A u t(X) denote its vertex set, edge set and automorphism group, respectively. Let G be a finite group and S a subset of G not containing the identity 1 G . The Cayley graph X = C a y(G,S) of G with respect to S is a graph defined by V(X) = G, E(X) = {(g,s g)|g ∈ G,s ∈ S}. In particular, if S−1 = S, such a graph can be viewed as an undirected graph by coalescing each pair, (g,s g) and (s g,g), of directed edges into a single undirected edge {g,s g}. A Cayley graph X = C a y(G,S) is called normal for G if the right regular representation of G is a normal subgroup of the automorphism group of X (see[1]). A graph X is called arc transitive or symmetric if A u t(X) acts transitively on the arc set of X.
Let X and Y be two graphs. The direct product X × Y is defined as the graph with vertex set V(X × Y) = V(X) × V(Y) such that, for any two vertices, u = (x1,y1) and v = (u2,v2) in V(X × Y), (u,v) is an edge in X × Y whenever x1 = x2 and (y1,y2) ∈ E(Y) or y1 = y2 and (x1,x2) ∈ E(X). The graphs are called relatively prime if they have no non-trivial common direct factor. The lexicographic product X[Y] is defined as the graph with vertex set V(X × Y) = V(X) × V(Y) such that, for any two vertices, u = (x1,x2) and v = (v1,v2) in V(X[Y]), (u,v) in an edge in X[Y] whenever (x1,x2) ∈ E(X) or x1 = x2 and (y1,y2) ∈ E(Y).
The concept of normality of the Cayley graph is known to be of fundamental importance for the study of arc transitive graphs. So, for a given finite group G, a natural problem is to determine all the normal or non-normal Cayley graph of G. Some meaningful results in this direction, especially for the undirected Cayley graphs, have been obtained. Baik et al.[2] determined all non-normal Cayley graphs of abelian groups with valency at most 4 and later[3] dealt with valency 5. For directed Cayley graphs, Xu et al.,[4] determined all non-normal Cayley graphs of abelian groups with valency at most 3.
An approach to analysing the family of Cayley graphs for a finite group G is given, which identifies normal edge transitive Cayley graphs as a subfamily of central importance. These are the Cayley graphs for G for which a subgroup of automorphisms exists, which both normalises G and acts transitively on edges. It is shown that, for a non-trivial group G, each normal edge transitive Cayley graph for G has at least one homomorphic image which is a normal edge transitive Cayley graph for a characteristically simple quotient group of G. For example, Alaeiyan et al.[5] determined all normal edge transitive undirected connected Cayley graphs of abelian groups with valancy at most 5 which are not normal. Our main result is as follows, the proof of which will be given in section ‘The proof of 1.1’.
Theorem 1.1
Let X = C a y(G,S) be a connected directed Cayley graph of an abelian group G with respect to S, and the size of S is at most 3. Then X is normal edge transitive if one of the following cases happens:
-
(1)
G = Z 2n = 〈a〉 (n > 2,n = 2k ),s = {a,a n + 1}, X = C n [2K 1]
-
(2)
G = Z n × Z 2 = 〈a〉 × 〈u〉 (n > 2,n = 2k), S = {a,a u},X = C n [2K 1]
The following corollaries are immediate consequences of the theorem.
Corollary 1.2
All non-normal connected Cayley graphs with valency 3 of a finite abelian group are not normal edge transitive.
Corollary 1.3
A non-normal connected Cayley graph with valency 2 of a finite abelian group is normal edge transitive if and only if o(G) = 4k.
Preliminary and result
For a graph X, we denote the automorphism group of X by A u t(X). The following propositions are basic.
Proposition 2.1
[6] Let X = C a y(G,S) be a Cayley graph of group G relative on S.
-
(1)
A u t(X) contains the right regular permutation of G, so X is vertex transitive.
-
(2)
X is connected if and only if G = < S >.
-
(3)
X is undirected if and only if S −1 = S.
Proposition 2.2
[7] Let Γ = C a y(G,S) be a Cayley graph for a finite group G with S ≠ ϕ. Then Γ is normal edge transitive if and only if A u t(G,S) is transitive on S, and if Γ is undirected, then Γ is normal edge transitive as an undirected graph if and only if A u t(G,S) is either transitive on S or has two orbits in S which are inverses of each other.
Proposition 2.3
[4] Let X = C a y(G,S) be a connected directed Cayley graph of an abelian group G with respect to S, and the valency of S at most 3. Then X is normal except when one of the following cases happens:
-
(1)
G = Z 2n = 〈a〉 (n > 2),S = {a,a n + 1},X = C n [2K 1]
-
(2)
G = Z n × Z 2 = 〈a〉 × 〈u〉 (n > 2),S = {a,a u}, X = C n [2K 1]
-
(3)
G = Z 4 = 〈a〉,S = G∖{1},X = K 4
-
(4)
G = Z 6 = 〈a〉,S = {a,a 3,a 5},X = K 3,3
-
(5)
G = Z 4 × Z 2 = 〈a〉 × 〈b〉,S = {a,a −1,b},X = Q 3
-
(6)
G = Z 2n × Z m = 〈a〉 × 〈c〉 (n > 2,m > 1), S = {a,a n + 1,c},X = C n [2K 1] × C m
-
(7)
G = Z n × Z 2 × Z m = 〈a〉 × 〈u〉 × 〈c〉 (n > 2, m > 1),S = {a,a u,c},X = C n [2K 1 × C m ]
-
(8)
G = Z 2n = 〈a〉,(n > 2),S = {a,a n + 1,a n}
-
(9)
G = Z n × Z 2 = 〈a〉 × 〈u〉 (n > 2),S = {a,a u,u}
-
(10)
G = Z 2k × Z 2 = 〈a〉 × 〈u〉 (k > 2),S = {a,a u,a k}
-
(11)
G = Z 2k × Z 2 = 〈a〉 × 〈u〉 (k > 2),S = {a,a u,a k u}
-
(12)
G = Z 4n = 〈a〉(n=4k + 1,k > 0), S = {a,a 2n + 1,a n + 1}
-
(13)
G = Z 4n × Z 2 = 〈x〉 × 〈y〉 (n = 2k + 1,k > 0), S = {x,x 2n + 1,x n + 1 y}
-
(14)
G = Z n × Z 4 = 〈a〉 × 〈u〉 (n = 4k,k > 0), S = {a,a v 2,a v}
-
(15)
G = Z k × Z t = 〈x〉 × 〈y〉,S = {x k / nh y,x k / nh y u,x k / mh y −1},u = (x k / nh y)nh / 2
-
(16)
G = Z k × Z t × Z 2 = 〈x〉 × 〈y〉 × 〈u〉, S = {x k / nh y,x k / nh y u,x k / mh y −1}
In both (15) and (16), k = m n h / (m,n) and t = (m,n). In (15), m is a positive integer; h > 1, 2 is not a divided h; and 2|n, n > 2 when n / 2 is odd, and n > 4 otherwise. In (16), m is a positive integer, h > 1, and n > 2.
The proof of 1.1
Let G be a finite abelian group, X = C a y(G,S) a connected directed Cayley graph of G with respect to S with valency at most 3. In this section, ‘The proof of 1.1’ will be completed by a series of lemmas. We will apply Proposition 2.2.
Lemma 3.1
The graphs X in cases (1) (for n odd), (2) (for n odd), (3), (4), (5), (6) (2n ≠ m), (7) (n ≠ m and n = m = 2k + 1), (8), (9), (11), (10), (12) and (13) in Proposition 2.3 are not normal edge transitive.
Proof
In case (1) (for n odd), (n + 1,2n) = 2r and a is a generator for G, thus there is no automorphism which takes a to an + 1 which means that A u t(G) cannot work transitively on S. In case (12), (n + 1,n) ≠ 1, similarly it is not normal edge transitive.
In case (2) (for n odd), O(a) ≠ O(a u), so there is no automorphism which takes a to au. In case (3), O(a2) = 2 and O(a) = 4. In case (4), O(a5) = 6 and O(a3) = 2. Thus, there is no automorphism which takes a to a2. In case (5), O(a) = 4 and O(b) = 2. In case (6) (for n ≠ 2m), O(a) ≠ O(c). In case (7) (for m ≠ n), O(a) ≠ O(c), and (for m = n = 2k + 1), O(a) ≠ O(a u). In case (8), O(a) ≠ O(an). In case (9), O(a) ≠ O(u). In case (10), O(a) ≠ O(ak). In case (11), O(a) ≠ O(aku). In case (13), O(xn + 1y) ≠ O(x). □
Lemma 3.2
The graphs X in cases (1) (for n even) and (2) (for n even) in Proposition 2.3 are normal edge transitive.
Proof
In case (1) (for n even), (n,n + 1) = 1, and since a is a generator of G, there is an automorphism which takes a to an + 1. It means that A u t(G,S) acts transitively on S.
In case (2) (for n even and r ∈ Z), define ϕ by ϕ(a2r) = a2r, ϕ(a2r + 1) = a2r + 1u, ϕ(a2ru) = a2ru and ϕ(a2r + 1u) = a2r + 1. Obviously, ϕ ∈ A u t(G,S), and so A u t(G,S) acts transitive on S and G is normal edge transitive. □
Lemma 3.3
The graphs X in cases (6) (2n = m), (7) (m = n = 2k) and (14) in Proposition 2.3 are not normal edge transitive.
Proof
For case (6) (2n = m), let ϕ ∈ A u t(G,S) and ϕ(a) = c, then ϕ(an + 1) = cn + 1 but it has to take an + 1 to a or an + 1. □
Similarly, for cases (7) (m = n = 2k) and (14), there are no automorphism to take a to c and a v2 to av, respectively.
Lemma 3.4
The graphs X in cases (15) and (16) in Proposition 2.3 are normal edge transitive.
Proof
In case (15), if (m,n) = 1 then S = {xmy,xmy(xmy)(nh)/2,xny−1}, clearly O(xmy) ≠ O(xmy−1), and so it is not normal edge transitive. □
Now let (m,n) = l. We have n = l r and m = l s, so xmy = xsy and xny−1 = xry−1. Clearly, O(xsy) ≠ O(xry−1).
Case (16) is similar to (15).
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MA and JM carried out the algebraic graph theory studies together and both of them participated in each part of this paper. Both authors read and approved the final manuscript.
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Alaeiyan, M., Mirzajani, J. Non-normal edge-transitive directed Cayley graphs of abelian groups. Math Sci 7, 7 (2013). https://doi.org/10.1186/2251-7456-7-7
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DOI: https://doi.org/10.1186/2251-7456-7-7