1. Introduction
All graphs in this paper are assumed to be finite, simple, connected and undirected.
Let
be a graph and denote
,
,
and
the vertex set, edge set, arc set and full automorphism group respectively. Denote
the valency of
. Then
is said to be X-vertex-transitive, Xedge-transitive and X-arc-transitive if
acts transitively on
,
, and
respectively. And further
is simply called vertex-transitive, edgetransitive and arc-transitive when
. Sometimes arc-transitive graph is simply called symmetric graph.
A graph
is a Cayley graph of a group
if there is a subset
with
such that
and
.
Bydefinition,
has valency
, and it is connected if and only if
. Moreover,
can be viewed as a regular subgroup of
by right multiplication action on V(G). For convenience, we still denote this regular subgroup by G. Then a Cayley graph is vertextransitive. On the contrary a vertex-transitive graph
is a Cayley graph of a group G if and only if
contains a subgroup that is regular on
and isomorphic to G (see [1, Proposition 16.3]). If G is a normal subgroup of
, then
is called a normal Cayley graph of G. The
is said to be core-free (with respect to G) if G is core-free in some
that is,
.
Let X be an arbitrary finite group with a core-free subgroup H and let D be a union of several double cosets of
satisfying
. The coset graph
is the graph with vertex set
![](https://www.scirp.org/html/8-1200119\bfd7db0e-ce78-490e-9e13-130b26ca0aa9.jpg)
such that
and
are adjacent if and only if
. Consider the action of X on
by right multiplication on right cosets. Note this action is faithful and preserves the adjacency of the coset graph, thus we identify X with a subgroup of
. Obviously,
is connected if and only if
. The valency of
is
. Let
be the set of vertices of
, which are adjacent with H. It is easy to check that H has n orbits on
if and only if D is the union of n double cosets of H. Further, the properties stated in the following lemma are well-known, its proof can be found in [2-4].
Proposition 1.1 Let
be defined as above.
1) If
is a X-symmetric graph of valency at least 3, then there exists an element
satisfying
and
. Furthermore, we may choose g to be a 2-element;
2) Let
be a Cayley graph and
. Let
be the stabilizer of
in X. We have
;
3) Let
be a coset graph and G be a complement of H in X. Denote
. Then the Cayley graph
is isomorphic to
, and hence
. In particular, S contains an involution of G if the valency of
is odd.
Tutte [5,6] proved that every finite connected cubic symmetric graph is
-regular for some
. Since Tutte’s seminal work, the study of s-arc-transitive graphs, aiming at constructing and characterizing such graphs, has received considerable attention in the literature, see [7-12] for example, and now there is an extensive body of knowledge on such graphs. Fang, Li, Wang and Xu [13] proved that for most finite nonabelian simple groups, the corresponding connected cubic Cayley graphs are normal. Caiheng Li [14] and Shangjin Xu [15] proved that every cubic symmetric Cayley graph of finite nonabelian simple group is normal except two 5-arc transitive graphs of the alternating group
(up to isomorphic). Then it arises a natural problem: whether each of the cubic non-symmetric Cayley graph of finite nonabelian simple group is normal? This problem has become the topics of greatest concern after the results of Li and Xu. Based on past experience, people conjure that if there exist some normal graphs, then the Cayley subsets of them must be consist of involutions. However, there have no any answer to the problem until now.
To answer this problem, by studying cubic nonsymmetric Cayley graphs, we give a negative answer. In the present paper, we give two non-normal examples which subsets are not consist of involutions. It’s worth noting that these are the first examples for the non-normal cubic nonsymmetric Cayley graphs of finite nonabelian simple groups.
In the rest of this section, we assume that
is a cubic nonsymmetric Cayley graph with
. Denote H the vertex stabilizer of X on
. Note
is cubic nonsymmetric, then H must be 2-group. Let N be the maximal one among normal subgroups of X contained in G, that is,
is the core of G in X.
2. Main Results
In this section, we construct some examples of cubic nonnormal nonsymmetric Cayley graphs on finite nonableian simple groups.
Example 2.1 Let
be the alternating group A15 and set
, where
![](https://www.scirp.org/html/8-1200119\2baa1b9b-9931-45d9-94cc-5f2387f3aa46.jpg)
Let
, then
is cubic nonsymmetric connected Cayley graph, which is not normal.
Let
be a connected graph, where
and
with
![](https://www.scirp.org/html/8-1200119\af87abc2-b9bb-4870-bbf3-8cd018e3a236.jpg)
It is easy to see
with 1 and 5 being in different orbits, which follows
. On the other hand,
and
lead to
.
Simple computation shows
and
, i.e.,
. Then the valency of
is
, and moreover
is nonsymmetric since
. Since
is connected, so
.
Let
. Clearly
acts transitively on
, which follows X acts 2-transitively on
and hence primitively, on
. Let
. Then
and X contains a 5-cycle
. Noting that every generator of X is even permutation,
by [16, Theorem 3.3E]. Then the stabilizer
. On the other hand H acts regularly on
leads to that
acts regularly on
. Simple computation shows
. Hence
by Proposition 1.1, and furthermore
is connected by the connectivity of
. Namely
, which leads to
. However
, which changes 1. Thus
, i.e., G is not normal in X.
Example 2.2 Let G be the alternating group
.
Set
, where
![](https://www.scirp.org/html/8-1200119\6f320f88-35f6-4656-b00f-a697cac6101d.jpg)
Let
, then
is cubic nonsymmetric connected Cayley graph, which is not normal.
Let
be a connect vertex-transitive graph, where
and
with
![](https://www.scirp.org/html/8-1200119\a7e2e279-0a5c-4221-88c9-ef87f49e2a4a.jpg)
It is trivial for us to get
with 1, 5 and 9 being in different orbits, then
. By simple checking, we find
and
. It follows
.
Note that
and
, then
. Namely the valency of
is
. However
, thus
is nonsymmetric. Notice that
is connected, i.e.,
.
Set
. Clearly
acts transitively on
, and then X acts 2-transitively on
, and hence primitively, on
. Let
.
Then
and X contains a 17-cycle
.
Note that all generators of X are even permutations, then
by [16, Theorem 3.3E]. Then the stabilizer
. Noting H acts regularly on
,
acts regularly on
. It is shown, by computing, that
. That is
by Proposition 1.1. And moreover the connectivity of
leads to
is also connected. Hence
, i.e.,
. However
![](https://www.scirp.org/html/8-1200119\a49eb738-09f0-4363-8f41-fb4a69a59921.jpg)
which changes 1. Thus
, i.e.,
is not normal in X.
NOTES
#Corresponding author.