On finite dual Cayley graphs

Abstract A Cayley graph Γ \Gamma on a group G is called a dual Cayley graph on G if the left regular representation of G is a subgroup of the automorphism group of Γ \Gamma (note that the right regular representation of G is always an automorphism group of Γ \Gamma ). In this article, we study finite dual Cayley graphs regarding identification, construction, transitivity and such graphs with automorphism groups as small as possible. A few problems worth further research are also proposed.


Introduction
Graphs considered in this article are finite and undirected. For a graph Γ , we denote by VΓ and Γ Aut its vertex set and its full automorphism group, respectively. For a vertex α of Γ , we denote by ( ) ≔ { ∈ | ∼ } Γ α β VΓ β α the neighbor set of α in Γ , where ∼ β α means that β is adjacent to α. , f o r a l l , ,ˇˇ: , f o ra l l , , 1 be the right regular representation and left regular representation of G, respectively. It is well known (also easy to prove) that Ĝ is always an automorphism group of Γ , but Ǧ is not necessarily. We call ( ) G S Cay , a dual Cayley graph on G if ≤ G Γ Aut . It is then natural and interesting to propose the following problem. Clearly, if G is abelian, then = G Ĝˇ, so the family of dual Cayley graphs on abelian groups is exactly the family of Cayley graphs on abelian groups. We thus mainly focus on nontrivial dual Cayley graphs, namely, dual Cayley graphs on nonabelian groups.
The aim of this article is to investigate the identification, construction and transitivity of dual Cayley graphs and propose several problems needing further research.

Properties of dual Cayley graphs
For a Cayley graph Γ on a group G, the following two notations will be used throughout the article: 1: the vertex of Γ corresponding to the identity element of G; τ: the permutation on VΓ via → ∈ − τ g g g G : f o r 1 . Other notations used in this article are standard. For example, for a positive integer n, we use n and D n 2 to denote the cyclic group of order n and the dihedral group of order n 2 , respectively, and use A n and S n to denote the alternating group and the symmetric group of degree n, respectively. Given two groups N and H, by N × H we denote the direct product of N and H, by N H . an extension of N by H, and if such an extension is split, then we write N H and Γ Aut . Furthermore, the following nice property holds.
The following theorem gives necessary and sufficient conditions for a Cayley graph to be a dual Cayley graph, where is the conjugate representation of G.
, h fixes S setwise. Then, we have the following equivalences.
ΓˆAut . Hence, ≤ G Γ Aut , and Γ is a dual Cayley graph on G. □ Denote by ( ) Z G the center of a group G. (ii) Proof.
x h x g ĥˇ1 for each ∈ x G, and by choosing = x 1 we obtain (ii) Obviously, ( ) = o τ 1 if G is an elementary abelian 2-group, and ( ) = o τ 2 otherwise. For each ∈ g G, as discussed above, For an element ∈ g G, denote by g G the conjugate class of g in G. The next lemma reveals the structure of dual Cayley graphs.
G S Cay , be a dual Cayley graph on a group G. Then, or equivalently, = − S S 1 is a union of some G-conjugate classes. Conversely, each Cayley graph constructed as above is a dual Cayley graph.
There are some typical symmetric graphs: K n (the complete graph with n vertices), K n n , (the complete bipartite graph with 2n vertices) and − K nK n n , 2 (the graph deleted a 1-match from K n n , ). It is known that all of them are Cayley graphs, but the next lemma shows not all of them are nontrivial dual Cayley graphs. Proof. Write = Γ K n , = Σ K n n , and = − Δ K pK p p , \ 1 1 is a union of some G-conjugate classes, by Lemma 2.4, K n is a nontrivial dual Cayley graph on G.
2 be a dihedral group of order n 2 . It is easy to show that . It then follows from Lemma 2.4 that K n n , is a nontrivial dual Cayley graph on H.
as p is an odd prime, a simple computation shows For a graph Γ, if a subgroup ≤ X Γ Aut acts transitively on the edge set or arc set of Γ, then Γ is called X-edge-transitive or X-arc-transitive, respectively. Both edge-transitive graphs and arc-transitive graphs are the main research objects in the field of algebraic graph theory. An arc-transitive graph is obviously edgetransitive. It is known that an edge-transitive Cayley graph In general, an edge-transitive Cayley graph is not necessarily arc-transitive. However, for dual Cayley graphs, edge-transitivity and arc-transitivity are equivalent, which was first observed in [5, Corollary 2.5], but the statement there is not precise.
be a dual Cayley graph on a nonabelian group G. Then, the following statements hold.
, -arc-transitive. In particular, each edgetransitive dual Cayley graph is arc-transitive. , namely, τ is an inner automorphism of G induced by g, which is a contradiction as G is a nonabelian group.

From Lemma 2.6, we have the next corollary. Recall that a Cayley graph Γ on a group G is called a normal
Cayley graph if ⊲ G Γ Aut , see [6].
Corollary 2.7. The dual Cayley graphs on nonabelian groups G are never normal Cayley graphs on G.

Proof. Suppose for a contradiction that Γ is a dual Cayley graph and is normal on a nonabelian group G.
Then, ⊲ G Γ Aut and by Theorem 2.2, ∈ τ Γ Aut ; however, as G is nonabelian, by Lemma 2.3, = ≠ G G Ĝˇτ , yielding a contradiction. has its full automorphism group as small as possible). There are many studies regarding graphical regular representations in the literature, which culminates in [7,8] with the classification of finite groups admitting GRRs. Motivated by Lemma 2.6(ii), we call that a dual Cayley graph ( ) G S Cay , with G nonabelian is a dual graphical representation with smallest automorphism group (DGRSA) of G if . It is then natural to ask the following basic "DGRSA" problem.
The following observation limits the structure of DGRSAs that are arc-transitive. for some ∈ { } g G\ 1 .
Proof. By assumption, by Lemma 2.3. By the arc-transitivity, the vertex stabilizer ( The next lemma shows that, for dihedral groups, only D 6 admits a unique arc-transitive DGRSA. for some ∈ g G, and by the connectivity, 〈 〉 = g G G . Set = g ab i j with ≤ ≤ − i n 0 1 and ≤ ≤ j 0 1. If = j 0, then ∈ 〈 〉 g a , and as 〈 〉 a is a characteristic subgroup of G, we deduce 〈 〉 ≤ 〈 〉 < S a G, a contradiction. Therefore, = g ab i is of order 2 and = S g G . Assume n is even.
: 〈 〉 ≅ ab D n , also a contradiction. Therefore, n is odd. Then, it is easy to see that each involution of G is conjugate to b, and a direct computation is of size n. Note that Γ is a bipartite graph with bipartitions 〈 〉 a and S, and we further conclude that ≅ Γ K n n , and = ( × ) Γ S S Aut : , which implies that = n 3 and ≅ Γ K 3,3 is an arc-transitive DGRSA of D 6 . □ We remark that, quite different from Lemma 2.9, Theorem 3.3 shows that each alternating simple group A n has an arc-transitive DGRSA, and Theorem 3.5 shows that ( ) p PSL 2, with ≥ p 5 a prime has an arc-transitive DGRSA if and only if ≡ ( ) p 1 mod 4 .  (1949) shows that cubic graphs are at most 5-arc-transitive, and Tutte's result was generalized by Weiss [9], which says that graphs except cycles are at most 7-arc-transitive. It is known that the complete graph K n with ≥ n 4 is 2-transitive, and the complete bipartite graph is 3-transitive.

Dual Cayley graphs on nonabelian simple groups
The following theorem classifies arc-transitive nontrivial dual Cayley graphs on nonabelian simple groups. Note that according to the structure and the action of the socle, the well-known O'Nan-Scott theorem divided the finite primitive permutation groups into eight types, see Praeger [10] or [11]. for some ∈ { } g T\ 1 . Furthermore, if Γ is 2-arc-transitive, as A is primitive on VΓ , by [12,Theorem 2], A should be an affine group, an almost simple group, or is of twisted wreath product type or of product action type, which is a contradiction as A is of simple diagonal type. Therefore, Γ is 1-transitive. □ Theorem 3.1 Particularly says that dual Cayley graphs on nonabelian simple groups are at most 2-transitive, and Lemma 2.9 shows that there is a 3-transitive nontrivial dual Cayley graph on soluble group. We have not found examples with higher transitivity on insoluble and nonabelian soluble groups.