Normal Edge-Transitive Cayley Graphs of the Group

Abstract

A Cayley graph of a group is called normal edge-transitive if the normalizer of the right representation of the group in the automorphism of the Cayley graph acts transitively on the set of edges of the graph. In this paper, we determine all connected normal edge-transitive Cayley graphs of the group .

1. Introduction

Let be a group and let be a subset of . The Cayley graph of with respect to is a graph with the group as the vertex set and is an arc of the graph, if and only if for some we have . We denote the Cayley graph by . Since is a group, the Cayley graph cannot have any parallel edges. In addition we assume that the subset does not contain the identity element of the group to avoid having loops in the Cayley graph and also to be an inverse closed set; that is, , to have an undirected Cayley graph. Therefore, we focus on simple Cayley graphs.

Let be a simple graph with vertex set and edge set . In this paper, we denote the edge, joining the vertices and by . An automorphism of the graph is a permutation on the vertex set of which preserves the edges. The set of all automorphisms of forms a group with the composition of maps as the binary operation and is denoted by . Since is a permutation group of , it acts on as well as in the usual way. is called vertex-transitive, edge-transitive, or arc-transitive, iff acts transitively on the set of vertices, edges, or arcs of , respectively. is called half-arc-transitive iff it is vertex- and edge-transitive, but not arc-transitive.

Let be a finite group and let be an inverse closed subset of that does not contain the identity. Set . is connected iff is a generating set of the group . For , define the mapping by , for all . for every , and hence is a regular subgroup of isomorphic to , forcing to be a vertex-transitive graph.

Let . Normalizer of in is equal to where denotes the semidirect product of two groups [1].

In [2], the graph is called normal if is a normal subgroup of . And after that the normality of Cayley graphs has been extensively studied from different points of view. Among them, finding and classifying the normal Cayley graphs were an essential problem, since in normal Cayley graphs we know the exact automorphism group of the graph.

It was conjectured in [2] that most Cayley graphs are normal. For example, in [3] the authors determined all possible nonnormal (and, as a consequence, normal) Cayley graphs of groups of order , and lots of other authors have done similar works for groups of orders , , , , and [4–8].

Another concept which was similar to the above one is introduced by Praeger in [9] in which a Cayley graph of a group with respect to is called normal edge-transitive or arc-transitive if acts on the edges or arcs of  , respectively, and it is called normal half-arc-transitive Cayley graph if it is normal edge-transitive Cayley graph which is not normal arc-transitive. Obviously, any normal edge-transitive Cayley graph is edge-transitive. Thus, this concept talks about the symmetric properties of a Cayley graph.

The latter concepts also were considered very much in the literature. For example, Alaeiyan in [10] found a class of normal edge-transitive Cayley graphs of abelian groups or, in [11], authors found all normal edge-transitive Cayley graphs of order and as a consequence found a class of normal half-arc-transitive Cayley graphs which rarely happens. This motivated us to consider the Cayley graphs of groups and classify all normal edge-transitive Cayley graphs of groups .

2. Preliminary Results

One of the principle theorems that helps us to connect the group properties of a group and normal edge-transitivity of is the following theorem which is proved in [9].

Theorem 1. Let be a connected Cayley graph on . Then, is normal edge-transitive if and only if is either transitive on or has two orbits in in the form of   and , where is a nonempty subset of such that .

The following corollary comes from Theorem 1 which is also mentioned in [11].

Corollary 2. Let and let be the subset of all involutions of the group . If does not generate the group and is connected normal edge-transitive, then the valency of is even.

Theorem 1 with a result in [12] characterizes arc-transitive and half-arc-transitive Cayley graphs as described in the following.

Theorem 3. Let be a connected normal edge-transitive Cayley graph. is normal arc-transitive if acts transitively on and is normal half-arc-transitive if , where is an orbit of the action of on .

3. Group

Group has the presentation which we can write its elements as follows: One can see that the group has order .

In Theorem 1, there are some relations between the normal edge-transitive Cayley graphs and the automorphism group of the relying group. Therefore, first we will find the automorphism group of the group .

Theorem 4. Every automorphism of the group is in the form of which sends to and to , or sends to and to , where for and ,  .

Proof. First of all we find some relations between elements of the group .
Since , equivalently Hence, we have One can check by induction that, for and , we will have If we denote the order of element of by , then, for , we have
For finding the order of elements of the form for , we have to observe that, for odd and arbitrary , we have and and, for even and arbitrary , we have , , and which can be obtained from (6).
Therefore, if the order of is , for odd , then for some integer . Since order of is , we should have But should be the least integer satisfying the condition and implies ; that is, where is the greatest common divisor.
Similar argument can apply for in the case is odd to obtain
Now for even , assume for some integer and for some integer . Thus, we have Compare it with the order of ; we get Since , one can conclude that , where is the least common multiple, and finally we obtain Similar argument can be discussed to prove Suppose is an automorphism of the group ; thus, preserves the order of elements; hence, and .
By the order of elements of  , if we define the following sets by the order of elements of the group , we can say that and . Let and . But all of the following cases which may happen for and yield a contradiction. (1) and contradicts with the fact that is a generating set of .(2) and . The equation implies which also implies or . But none of them can occur, since .(3) and . Thus, for some odd and for some . Hence, is even and is odd and we have But because is an automorphism and hence yields , which is a contradiction.(4) and . We can write , for odd , and , for and . Thus, and, hence, are odd and we obtain But we have . Thus, we should have ; that is, , which is a contradiction.Therefore, the only cases that may happen are the cases , and , and the theorem is proved.

And from Theorem 4, we can obtain that the automorphism group of the group is as follows:

Now, we are ready to find the orbits of the group under the action of .

Lemma 5. In the action of group on , each element falls into one of the following orbits, depending on : (I); (II); (III); (IV).

Proof. By Relation (6), one can show that, for odd , if is even, then , and if is odd, then for . Therefore, images of elements of under and in are as follows (recall that , ).
If is even, then
If is even or odd, then and . Therefore, the orbit of for an even is and for odd is . The orbit of for an even is and for odd is . And finally the orbit of is .

Now we bring a sufficient condition under which a Cayley graph of group can be normal edge-transitive.

Theorem 6. If is connected normal edge-transitive Cayley graph, then is even, greater than 2, and is contained in .

Proof. From Corollary 2, is even.
Since is normal edge-transitive, by Theorem 1, acts transitively on or where is an orbit of the action of . Since is a subgroup of , thus or, in the latter case, should be contained in one of orbits of on . But, from Lemma 5, we have 4 kinds of orbits in this action. But from (6), if and are even, . Thus, is even and is not generated by such elements. Therefore, if is contained in an orbit of kind I, II, or IV, will not generate . Therefore, will not be connected. Thus, is contained in III.

Theorem 6 has a critical corollary.

Corollary 7. is never connected normal half arc-transitive Cayley graph.

Proof. For odd and , we have . From Theorem 6, contains such elements. Since is closed under inversion, is union of for some odd and some . is an automorphism of which preserves , for all and , and hence preserves ; that is, . Accordingly, if we set , then each element of is sent to an element of by . is not an orbit of the action of on .

By Theorems 1 and 6 and Corollary 7 we can obtain the following theorem.

Theorem 8. is connected normal edge-transitive Cayley graph if is even, greater than 2, is contained in , and acts transitively on .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank anonymous referees for their useful comments and suggestions.