/^creative ^commor
ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 15 (2018) 407-424 https://doi.org/10.26493/1855-3974.1345.ae6
(Also available at http://amc-journal.eu)
Isomorphisms of generalized Cayley graphs*
Xu Yang
School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai, 201209, China, and School ofMathematics and Statistics, Central South University, Changsha, Hunan, 410083, China
Weijun Liu, Lihua Feng t
School ofMathematics and Statistics, Central South University, Changsha, Hunan, 410083, China
Received 5 March 2017, accepted 2 December 2017, published online 12 August 2018
Abstract
In this paper, we investigate the isomorphism problems of the generalized Cayley graphs, which are generalizations of the traditional Cayley graphs. We find that there are two types of natural isomorphisms for the generalized Cayley graphs. We also study the GCI-groups among the generalized Cayley graphs, and the Cayley regressions of some groups. We mainly showed that, for an odd prime power n, Z2n (resp. D2n) is a restricted GCI-group if D2n (resp. Z2n) is a CI-group. We also obtain that the cyclic group of order 2n is a 4-quasi-Cayley regression if and only if n = 3.
Keywords: Generalized Cayley graph, natural isomorphism, GCI-group, Cayley regression. Math. Subj. Class.: 05C25, 20D20
1 Introduction
Let G be a finite group, S C G be a subset and a e Aut(G). If G, S, a satisfy the following three conditions:
*The authors would like to express their sincere thanks the referees for their valuable comments, corrections and suggestions which lead to a great improvement of this paper. L. Feng, as the corresponding author, would like to thank SDIBT for their hospitality. This work was supported by NSFC (Nos. 11671402, 11271208, 11371207), Hunan Provincial Natural Science Foundation (2016JJ2138, 2018JJ2479), Mathematics and Interdisciplinary Sciences Project of CSU. t Corresponding author.
E-mail address: xcubicy@163.com (Xu Yang), wjliu6210@126.com (Weijun Liu), fenglh@163.com (Lihua Feng)
©® This work is licensed under https://creativecommons.org/licenses/by/3.0/
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(i)	a2 = 1;
(ii)	if g € G, then a(g-1)g € S;
(iii)	if g, h € G and a(g-1)h € S, then a(h-1)g € S,
then the structure r = GC(G, S, a) is called a generalized Cayley graph with V(r) = G, E(r) = {{g, h} | a(g-1)h € S}. The neighborhood of a vertex g € G is the set of vertices adjacent to g, denoted by N(g). Then N(g) = {a(g)s | s € S}.
According to condition (i), a is either the identity of Aut(G) or an involution. When a is the identity, then the definition of GC(G, S, a) is just the same as that of Cayley graphs, and thus GC(G, S, a) = Cay(G, S). In this case, S is symmetrical, i.e., S = S-1 = {s-1 | s € S} and for a € Aut(G), we have that a acts on V(r) naturally as V(r) = G. Also, if T = Sff, then there is a bijection from r to r = Cay(G, T) induced by a, defined as a: V(r) ^ V(rff), g ^ ga. It follows r = r. This kind of isomorphism between Cayley graphs induced by the automorphisms of G is called the Cayley isomorphism. It should be mentioned that not all isomorphisms between Cayley graphs are Cayley isomorphisms. In fact, there are pairs of isomorphic Cayley graphs with no Cayley isomorphism between them. This encourages us to investigate the so-called Cl-graphs and Cl-groups defined below.
Definition 1.1. A Cayley graph Cay(G, S) is called a Cl-graph of G, if for any Cayley graph Cay(G, T), Cay(G, S) = Cay(G, T) implies S= T for some a € Aut(G). In this case, S is called a Cl-subset. Furthermore, G is called a Cl-group if any symmetrical subset not containing the identity is a CI-subset.
For those graphs having particular transitive properties, such as Cayley graphs and bi-Cayley graphs, their isomorphism problems are well studied in the literature (recall that a bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits on the vertices). The isomorphism problem for Cayley graphs was proposed decades ago and has been investigated deeply up to now. It was initiated by Adam in 1967 who conjectured that any cyclic group is a DCI-group, where a DCI-group satisfies that any subset not containing the identity and not necessarily symmetrical is a CI-subset. Although this conjecture was soon denied by Elspas and Turner [4], it stimulated the study of CI- and DCI-groups. Alspach, Parsons [1] and Babai [3] presented a criteria for CI-graphs. Muzy-chuk [18, 19] obtained a complete classification of the CI-groups in finite cyclic groups. Li [14] showed that all finite CI-groups are solvable. The isomorphism problem and the automorphism groups for bi-Cayley graphs have also been studied flourishingly; one may refer to [10, 11, 28]. Other related results could be found in [15, 16, 23, 24, 26, 27].
The concept of generalized Cayley graphs was introduced by Marusic et al. [17] when they dealt with the double covering of graphs. Answering a question in [17], the authors in [8] found some vertex-transitive generalized Cayley graphs which are not Cayley graphs. Further, the authors in [25] studied the isomorphism problems of generalized Cayley graphs and found that the alternating group An is a restricted GCI-group if and only if n = 4.
The present paper can be regarded as the continuance of the above work, and also provides support to the question at the end of [8], where the authors asked for the classification of all generalized Cayley graphs arising from cyclic groups. The structure of this paper is as follows. In Section 2, we give several properties of the generalized Cayely graphs and some lemmas which will be used later. In Section 3, we introduce two types of natural
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isomorphisms for any generalized Cayley graph. In Section 5, we study the GCI-groups in cyclic groups. We show that when G is a dihedral group of order 2n with n an odd prime power, if G is a Cl-group, then Z2n is a restricted GCI-group. In Section 6, we study the GCI-groups in dihedral groups. We show that when G is a cyclic group of order 2n with n an odd prime power, if G is a CI-group, then D2n is a restricted GCI-group. In Section 7, we study the Cayley regressions, a concept relating to both Cayley graphs and generalized Cayley graphs. We show that the cyclic group Z2n is a 4-quasi-Cayley regression if and only if n = 3. Finally, we propose some questions for future research.
2 Preliminaries
All graphs considered in the paper are simple, finite and undirected. All the automorphisms in the paper that induce generalized Cayley graphs are assumed to be some involutions.
Let G be a finite group that admits an automorphism a of order two. For g =1, we have a(h-1) G S whenever h G S, by condition (iii), implying a(S) = S-1. Let wa: G ^ G be the mapping defined by wa(g) = a(g-1)g for any g G G. Note that wa is not necessarily a bijection. Let wa(G) = {wa(g) | g G G}. We use the same notation and terminology as in [8]. Suppose s G S, then a(s) G a(S), and thus a(s) G S-1. Therefore s G S if and only if a(s-1) G S. Let Qa be the set containing all elements satisfying a(g) = g-1 in G \ wa(G), and be the set containing all elements in G satisfying a(g) = g-1. Let Ka = {g G G | a(g)g = 1}. Then we have
Proposition 2.1 ([25]). Let GC(G, S, a) be a generalized Cayley graph of G. Then
(1)	S n wa(G) = 0. Conversely, if S n wa(G) = 0, a is an involution in Aut(G) and a(S) = S-1, then G, S, a can induce a generalized Cayley graph.
(2)	G = Ka U and Ka = wa(G) U Qa. Furthermore, wa(G),	are all symmetrical.
(3)	S = S1 U S2, where S1 C and S2 C
Proposition 2.2. Let G be a finite group admitting two automorphisms a, ft of order two. If a, ft are conjugate in Aut(G), then Cay(G, wa(G) \ {1}) = Cay(G, w^ (G) \ {1}).
Proof. By Proposition 2.1, we have wa(G) = wa(G)-1 and (G) = (G)-1. Since a, ft are conjugate, there exists some 7 G Aut(G) such that ft = YaY-1 = aY. Therefore
7(wa(G)) = {Y(a(g-1)g) | g G G}
= {YaY-1Y(g-1)Y(g) 1 g G G} = {ft(Y(g)-1)Y(g) I Y(g) G G} = w^ (G).
It follows that Y(wa(G) \ {1}) = w^(G) \ {1}. Hence the result follows.	□
Theorem 2.3. Let G be a finite group admitting an automorphism a of order two, S C G such that S n wa(G) = 0. Let $(g) = a(g)Sg-1. If S is symmetrical and $(g) = S for any g G G, then GC(G, S, a) = Cay(G, S).
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Proof. Let r = GC(G, S, a) and r2 = Cay(G, S). Let ^: V(ri) ^ V(r2), x ^ x-1 be a bijection between these two graphs. For any {g, h} e E(r1), there exists some s e S such that h = a(g)s. {g, h}^ = {g-1, h-1}. Note that gh-1 = gs-1a(g)-1 = a(a(g))s-1a(g)-1. Since S is symmetrical and $(g) = S for any g e G, we have
a(a(g))s-1a(g)-1 e S. This implies {g, h}^ e E(^), and thus GC(G,S,a) = Cay(G,S).	□
Theorem 2.3 can be regarded as a criteria to judge whether some generalized Cayley graphs are Cayley graphs or not.
Theorem 2.4. Let G be any finite group admitting an automorphism a of order 2. Then we have GC(G, S, a) = Cay(G, S), where S = Ua, Qa or G \ wa(G).
Proof. By proposition 2.1, G = wa(G) U Qa U Ua. For any g e G, G = a(g)Gg-1. For any x e wa(G), there exists some h e G such that x = a(h-1)h. So a(g)xg-1 = a(g)a(h-1)hg-1 = a((hg-1)-1)hg-1, and hence wa(G) = a(g)wa(G)g-1. Asaresult, U = a(g)^ag-1 U a(g)^ag-1.
For any s e assume that a(g)sg-1 e then a(a(g)sg-1)-1 e and a(g)sg-1 = a(a(g)sg-1)-1. Since a(a(g)sg-1)-1 = a(g)a(s-1)g-1, we have that s = a(s-1), which is a contradiction as s e This means a(g)sg-1 e Thus, Qa = a(g)^ag-1 and = a(g)^ag-1. By Theorem 2.3, we get the result.	□
Let Fix(a) = {g e G | a(g) = g}. So Fix(a) < G and we have the following lemma. Lemma 2.5 ([8]). |wa(G)| = jFiaTl •
Note that some references also use CG(a) to denote Fix(a). Those papers mainly investigate the properties of the finite groups which admit involutory automorphisms; one can refer to [2, 13, 21, 22]. Although those problems are not considered in this paper, we borrow the following well-known result.
Lemma 2.6 ([7]). Let G be a finite group of odd order admitting an automorphism ^ of order two. Then the following statements hold.
(1)	G = FK = KF, F n K =1, and |K| = |G : F|, where F = CG(^) and K = K0;
(2)	Two elements of K conjugate in G are conjugate by an element of F;
(3)	If H is a subgroup of F, then NG(H) = CG(#)Nf(K).
By Lemmas 2.5 and 2.6, we get
Proposition 2.7. Let G be a group of odd order admitting an automorphism a of order two. Then Qa = 0.
Proof. By Lemmas 2.5 and 2.6, |Ka| = |wa(G)| = | ^G^). As Ka = wa(G) U we obtain tta = 0.	□
Remark 2.8. By Proposition 2.7, for any generalized Cayley graph GC(G, S, a), if |G| is odd, S C Ua. We present an alternative proof avoiding Lemmas 2.5 and 2.6. If Qa = 0, assume that = 0. Then G is an abelian group of odd order by Proposition 2.1. Thus a is a fixed-point-free automorphism of G. Then Ka = wa(G) = G according
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to [7, Lemma 10.1.1], which is a contradiction. This implies that Ua = 0. Since the S in GC(G, S, a) are choosen from Qa and Ua. Therefore |S| must be odd, which is a contradiction as, there are no regular graphs of odd order with odd valency. This implies
Qa = 0.
It is well known that a finite group G of odd order is solvable by Feit-Thompson Theorem [5]. From above, we can see that the classification of GC(G, S, a) of finite group G of odd order seems to be more clear as the elements of S can only be chosen from since
Qa = 0.
In [8], Hujdurovic et al. defined the following set
Aut(G, S, a) = jp G Aut(G) | p(S) = S, ap = pa}.
Moreover, one sees that Aut(G, S, a) = Aut(G, S) n CAut(G)(a), where Aut(G, S) = Aut(G, S, 1).
Proposition 2.9. Let S be the set as in (3) of Proposition 2.1. Then Aut(G, S, a) = Aut(G,Si,a) n Aut(G,S2,a) = Aut(G,Si) n Aut(G, S2) n CAut(G)(a). Furthermore, the couples of the form like js, a(s-1)} are imprimitive blocks of Aut(G, S, a).
Proof. For any s G S1 and s' G S2, if there exists some p G Aut(G, S, a) such that s = p(s'), then ap(s') = a(s). Since ap = pa and s = a(s-1), pa(s'-1) = s. This implies a(s') = s'-1, which is a contradiction as s' G S2. Hence p(S1) = S1 and p(S2) = S2 for any p G Aut(G, S, a).
Let A = js, a(s-1 )} be a couple in S2. For any p G Aut(G, S, a), Av Ç S2. If A n Av = 0, then s = p(s) or s = pa(s-1). If s = p(s), then a(s-1) = pa(s-1). If s = pa(s-1), then a(s-1) = p(s). This implies that A = Av. Thus A is an imprimitive block.	□
Let GC(G, S, a) be a generalized Cayley graph of G. Under the condition of Proposition 2.9, S n S-1 = (S1 u S2) n (S1 u S2)-1 = (S1 n S-1) u (S1 n S-1) u (S2 n S-1) U (S2 n S-1). Note that S1 n S-1 = S2 n S-1 = 0, it follows that S n S-1 = (S1 n S-1) U (S2 n S-1). Since S1 Ç and Qa is symmetrical, so S1 n S-1 Ç Similarly, S2 n S2-1 Ç Ua. Let T = S n S-1. It follows that GC(G,T,a) is still a generalized Cayley graph of G. We call GC(G, T, a) the induced generalized Cayley graph of GC(G, S, a). Note that T-1 = T, this encourages us to consider the Cayley graph Cay(G, T), called the induced Cayley graph of GC(G, S, a). Next we consider Aut(G, S, a), Aut(G, T, a) and Aut(G, T).
Proposition 2.10. Aut(G, S, a) < Aut(G,T, a) < Aut(G,T). Furthermore, Aut(G, S, a) < Aut(G, T, a) if S is not symmetrical; Aut(G, T, a) = Aut(G, T) if a G Z(Aut(G)).
Proof. For any p G Aut(G, S, a), we have p(S) = S and p(S-1) = S-1, thus p(T) = T, p G Aut(G, T, a). If S is not symmetrical, we have a G Aut(G, S, a) as a(S) = S-1 = S, but a G Aut(G, T, a) as a(T) = T. Aut(G, T, a) < Aut(G,T) is obvious by the definition. Since Aut(G, T, a) = Aut(G, T) n CAut(G)(a), we get the result.	□
Finally, we introduce a lemma about the connectivity of the generalized Cayley graph.
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Lemma 2.11. Let G be a group, A C G and a G Aut(G) of order 2. The generalized Cayley graph X = GC(G, A, a) is connected if and only if A is a left generating set for (G, *), where f * g = a(f )g for all f, g G G.
3 Two basic types of isomorphisms
In this section, we will introduce two types of natural isomorphisms of generalized Cayley graphs for any finite group. First, we introduce the first type of natural isomorphism found by A. Hujdurovic et al.
Theorem 3.1 ([9]). GC(G, S, a) = GC(G, S^, a^) for any fi G Aut(G), where
aP = fiafi-1.
Remark 3.2. From Theorem 3.1, one can see that if a, 7 are conjugate, then there is a generalized Cayley graph GC(G, S, a) if and only if there is a generalized Cayley graph GC(G, S^, y) with y = a/ such that these two graphs are isomorphic. Hence, if we intend to study all the generalized Cayley graphs of some group G, we only need to study the generalized Cayley graphs related to the representatives of the conjugacy classes of elements in Aut(G).
Corollary 3.3. GC(G, S, a) = GC(G, S-1, a).
Proof. Let fi = a. Then GC(G, S, a) = GC(G, a(S), aa) by Theorem 3.1. Note that a(S) = S-1, this completes the proof.	□
Next, we introduce the second type of natural isomorphism.
Theorem 3.4. Let GC(G, S, a) be a generalized Cayley graph. Then GC(G, a(g)Sg-1, a) is also a generalized Cayley graph of G for any g G G. Furthermore, GC(G, S, a) = GC(G, a(g)Sg-1, a).
Proof. For any x G G, if a(x-1)x G a(g)Sg-1, a(g-1)a(x-1 )xg G S, that is, a((xg)-1)xg G S, which conflicts with condition (ii). If a(x-1)y G a(g)Sg-1, then we have a((xg)-1)yg G S. Thus a((yg)-1)xg G S by condition (iii). It follows that a(y-1)x G a(g)Sg-1. Therefore, GC(G, a(g)Sg-1, a) is also a generalized Cayley graph of G for any g G G.
Let r = GC(G, S, a) and rg = GC(G, a(g)Sg-1, a). Let 0: V(r) ^ V(rg), a ^ ag-1. So 0 is abijection. For any {a, b} G E(r), a(a-1)6 G S. Since
a((ag-1)-1)(bg-1) = a(g)(a(a-1)b)g-1 G a(g)Sg-1,
we have {ag-1 ,bg-1} G E(rg). Therefore {a, b} G E(r) if and only if {a, 6}e G E (ra). Thus they are isomorphic.	□
According to Theorem 3.1, r = r^ for any fi G Aut(G), we call the mapping x ^ x^ the the first basic type of isomorphism of r. By Theorem 3.4, r = rg for any g G G, we call the mapping x ^ xg-1 the second basic type of isomorphism of r.
For any g G G, R(g): x ^ xg is a permutation of G. Set R(H) = {R(h) | S = a(h)Sh-1}.
Theorem 3.5. Let r = GC(G, S, a) be a generalized Cayley graph. Then R(H) < Aut(r).
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Proof. For any {a, b} G E(T), it suffices to show that {a, b}R(h) G E(r) for any R(h) G ). Since {a,b} G E(r), a(a-1)b G S = a(h)Sh-1. It follows that a((ah)-1)bh G S, which implies that {ah,bh} G E(r). Thus R(h) G Aut(r). For any R(h),R(h') G R(H), S = a(h)Sh-1 and S = a(h')Sh'-1. Therefore S = a(h/-1h)S(h/-1h)-1, thus R(h'-1h) G R(H). This implies that R(H) < Aut(r).	□
4 GCI, restricted GCI and strongly GCI groups
Similarly to the Cl-groups in Cayley graphs and BCI-groups in bi-Cayley graphs, we propose the following definitions relating to generalized Cayley graphs.
Definition 4.1. Let G be a finite group. Let M be the set of all Cayley graphs and N be the set of all generalized Cayley graphs constructed by automorphisms of order two. Then
1.	G is called a GCI-group if both of the following are satisfied:
(i)	for any two nontrivial generalized Cayley graphs GC(G, S, 1) and GC(G, T, 1) in M, whenever GC(G, S, 1) = GC(G, T, 1), there exists 5 G Aut(G) such that S5 = T.
(ii)	for any two nontrivial generalized Cayley graphs GC(G, S, a) and GC(G, T, P) in N, whenever GC(G, S, a) = GC(G, T, P), there exists 5 G Aut(G) such
that P = a5 = 5a5-1 and T = a5(g)S5 g-1.
2.	G is called a restricted GCI-group if (ii) is satisfied.
3.	G is called a strongly GCI-group if for any nontrivial GC(G, S, a), whenever
GC(G, S, a) = GC(G, T, P), there exists 5 G Aut(G) such that P = a5 = 5a5-1 and T = a5 (g)S5 g-1.
Remark 4.2.
1.	The definition is based on Theorems 3.1 and 3.4 and Definition 1.1. The two basic types of isomorphisms and their compositions are called the natural isomorphisms of generalized Cayley graphs. For instance, GC(G, S, a) = GC(G, SY, aY) by Theorem 3.1, GC(G, SY, aY) = GC(G, aY(g)SYg-1,aY) by Theorem 3.4, then we have GC(G, S, a) = GC(G, aY(g)SYg-1, aY).
2.	The word 'nontrivial' in the definition means that the null graph is not considered. In fact, if it is included, for a finite group G which has an automorphism a of order 2, GC(G, 0,1) and GC(G, 0, a) are both isomorphic to the null graph. By the definition, G cannot be a strongly GCI-group, otherwise it will make the definition meaningless, thus the null graph is not considered in the definition.
3.	If a finite group G has no automorphisms of order two, then we still consider that (ii) is satisfied for G.
4.	By definition, strongly GCI-group implies GCI-group, GCI-group implies CI-group and restricted GCI-group. However, restricted GCI does not imply GCI and does not imply CI either. If G is not a restricted GCI-group or a CI-group, then it is not a GCI-group either.
Next we will give some examples of finite groups satisfying special conditions:
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Example 4.3. Let G = Z4. Then G is a GCI group by Theorem 5.2. However, let a: x ^ —x be an involution. Thus GC(G, {1}, a) is a generalized Cayley graph of G. Also, GC(G, {2}, 1) is a generalized Cayley graph of G. Although GC(G, {1}, a) = GC(G, {2}, 1) but, a is not conjugate to 1, that means G is not a strongly GCI group. Therefore Z4 is a GCI but not strongly GCI group.
Let G = Z8. Then G is a CI group [19]. However, Z2n is a GCI group if and only if it is Z2 or Z4 by Theorem 5.2. It follows that G is not a GCI group. Thus Z8 is a CI but not GCI group.
Though we find example of CI but not restricted GCI groups, like Z8, we have not found out the example of restricted GCI but not CI groups up to now. Thus we propose the following question:
Question 4.4. Is every restricted GCI group a CI group?
The next theorem is useful to determine whether a group is a restricted GCI-group or
not.
Theorem 4.5. Let G be a finite group admitting two automorphisms a, ft of order two. If a, ft satisfy the following three conditions:
(1)	a and ft are not conjugate;
(2)	K(G)| = |Ka|;
(3)	(G)| = K |,
then G is not a restricted GCI-group.
Proof. Assume |G| = n. If these three conditions are satisfied, then n is even by Proposition 2.7. Furthermore, there must exist two generalized Cayley graphs, say GC(G, {s}, a) and GC(G, {s'}, a), which are both isomorphic to nK2. But there is no natural automorphism as a and ft are not conjugate. Hence G is not a restricted GCI-group.	□
To conclude, we give the characterization of strongly GCI-groups.
Theorem 4.6. A finite group G is a strongly GCI-groups if and only if G is a Cl-group and one of the following is true for G:
(1)	G has no involutory automorphisms;
(2)	all involutory automorphisms are fixed-point-free.
Proof. First we show the necessity. If G is a strongly GCI-groups, then G must be a CI-group. If not all involutory automorphisms of G are fixed-point-free automorphisms or, as we will show that G has no automorphisms of order two. If there exists some involutory automorphism which is not fixed-point-free, say a, this means | Fix(a) | = 1. By Lemma 2.5, we get wa(G) = G. Since G = wa(G) U Qa U by Proposition 2.1, it follows that Qa U = 0. Thus at least one of Qa and Ua, say is not an empty set. According to Theorem 2.4, GC(G, a) = GC(G, 1) which is not a null graph. This is a contradiction to the fact that G is a strongly GCI-group. Therefore G has no automorphisms of order two since otherwise all automorphisms of order two of G are fixed-point-free automorphisms. If G has no automorphisms of order two, then G must be a CI-group as G is a strongly GCI-group.
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Next we show the sufficiency. Suppose that all automorphisms of order two of G are fixed-point-free. Let a G Aut(G) be such an involution. Then G = wa(G) by Lemma 2.5, so any generalized Cayley graph induced by involutory automorphism is a null graph. □
5 The cyclic GCI groups
Theorem 5.1. The cyclic group of order pn with p an odd prime is a GCI-group if and only if it is a CI-group.
Proof. Let G = . Then G has only one automorphism of order two, that is a: x ^ —x. Note that Wa(G) = {a(g-1)g | g G G} = {2g | g G G}, it follows that S = 0 as any non-identity of G is a square since |G| is odd. Thus the only generalized Cayley graph of G induced by automorphisms of order two is GC(G, 0, a) = pnK1.	□
Babai [3] classified the Cl-groups of cyclic groups of order 2p with p a prime. God-sil [6] classified the Cl-groups of cyclic groups of order 4p. Next we will classify the GCI-groups of cyclic groups of even order. We will deal with the problem step by step in this section.
Theorem 5.2. Let G be a finite cyclic group of order 2n. Then G is a GCI-group if and only if n = 1, 2.
Proof. Let G = Z2n = {0,1,..., 2n — 1}. When n =1, Aut(G) = 1, there are no automorphisms of order two in Aut(G). Therefore G is a GCI-group by Definition 4.1. When n = 2, then Aut(G) = Z2, there is a unique element of order two in Aut(G) since Aut(G) is cyclic, say a: x ^ —x. If g G G, then a(g-1)g = 2g G S. Hence S C {1,3}. Therefore there are only three generalized Cayley graphs of G, with S being {1}, {3} and {1, 3}, respectively. Let r = GC(G, {1}, a), r2 = GC(G, {3}, a). Note that —1 = 3 (mod 4), and so r1 = r2 by Corollary 3.3.
When n > 3, then Aut(G) = Z2 x Z2n-2, and there are only three automorphisms of order two in Aut(G), say,
a: x ^ —x, ft: x ^ (2n-1 — 1)x, 7: x ^ (2n-1 + 1)x.
Let S = {1,2n-1 + 1}. Since 1 ^ 2n-1 + 1 (mod 2n) and they are both odd, we have S n wa(G) = 0 as wa(g) = a(g-1)g = 2g is even for any g G G. Further, S n wg (G) = 0 as wg (g) = ft(g-1)g = 2n-1g + 2g (mod 2n) is also even for any g G G. Recall that ft( —1) = 2n-1 + 1, a( —1) = 1 and a(—(2n-1 + 1)) = 2n-1 + 1, hence a(S) = S-1 and ft(S) = S-1. Therefore both GC(G, S, a) and GC(G, S, ft) are generalized Cayley graphs of G.
Let r = GC(G, S, a). Since |S| = 2, the valency of r is two. For any x G V(r1), N(x) = {a(x) + y | y G S} = {—x + 1, —x + 2n-1 + 1}. Consider the vertex 2n-1 + x (mod 2n), it follows that x ^ 2n-1+x (mod 2n). N(2n-1+x) = {a(2n-1 +x)+y | y G S} = {2n-1 — x +1, —x + 1}. Thus 'x ^ (—x +1) ^ (2n-1 + x) ^ (2n-1 — x + 1) ^ x' is a 4-cycle in r1. Therefore r1 = 2n-2C4.
Let r2 = GC(G, S, ft). Since |S| =2, the valency of r2 is two. For any x G V(^), N(x) = {ft(x) + y | y G S} = {(2n-1 — 1)x + 1, (2n-1 — 1)(x — 1)}. We consider the vertex 2n-1 + x (mod 2n). Then N(2n-1 + x) = {ft(2n-1 + x) + y | y G S} = {(2n-1 — 1)x — 2n-1 + 1, (2n-1 — 1)x +1}. Thus 'x ^ (2n-1 — 1)x +1 ^ (2n-1 + x) ^ (2n-1 — 1)(x — 1) ^ x' is a 4-cycle in r1. Therefore r2 = 2n-2C4.
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From above, GC(G, S, a) = GC(G, S, ft) = 2n 2C4, but a and ft are not conjugate in Aut(G) as Aut(G) is abelian, hence G is not a restricted GCI-group by Definition 4.1. □
Theorem 5.3. Let G be a finite cyclic group of order 2apb with p an odd prime and a, b > 0. If G is a restricted GCI-group, then a = 1.
Proof. Since G is a finite cyclic group of order 2apb, let G = Gi x G2, where Gi = Z2a
and G2 = Zpb.
We claim that a < 2. Now we suppose a > 3. By Theorem 5.2, a: (g1,g2) ^ (-g1,g2) and ft: (g1,g2) ^ ((2n-1 - 1)g1,g2) are two different automorphisms of G with order two when a > 3. Let S = {(1,0), (2n-1,0)}. Then GC(G,S,a) and GC(G, S, ft) are two generalized Cayley graphs of G. According to Theorem 5.2, we have GC(G, S, a) = GC(G, S, ft) = 2n-2p6C4. Note that a and ft are not conjugate in Aut(G), it follows that a < 2.
Assume a = 2. Note that a: (01,02) ^ (-51,52), ft: (01,02) ^ (01, -02) are two automorphisms of G. Furthermore, wa(G) = {(0,0), (2,0)}, Ka = {(01,0) | 01 G G1}. Therefore Qa = {(1,0), (3,0)}. wp(G) = {(0,02) | 02 G G2}, Kp = {(0,02), (2,02) | 02 G G2}. Thus Qp = {(2,02) | 02 G G2}. Let S1 = {(1,0)} and S2 = {(2,0)}. We can see that GC(G, S1, a) = GC(G, S2, ft) = 2pbK2. However a and ft are not conjugate as Aut(G) is abelian. It follows from above discussion that a =1.	□
Theorem 5.4. Let G be a finite cyclic group of order n, where n is even with at least two different odd prime divisors. Then G is not a restricted GCI-group.
Proof. Suppose that n = p00 • p11 • • • p6kk, where po = 2, pi,pj are different odd primes for any i, j G {1,..., k} and st > 1 is an integer for any t G {0,1,..., k}, k > 2. It follows that G can be decomposed into the direct product of some cyclic groups, say
G = G0 x • • • x Gk = Z2so x Zpsi x • • • x Zpsfc, where Gi = Z , i = 0,1,..., k.
p1
Let and
(xo ,x1,...,xk) ^ (xo, -x1,...,xk)
ft: (xo, X1, X2,... ,xk) ^ (xo,X1, -X2,..., xk).
Since k > 2, then such a, ft can not appear in Aut(G). Obviously wa(G) = {(0, x1,0,..., 0) | x1 G G1} and wp (G) = {(0,0,x2,..., 0) | x2 G G2}. Let 0i G Gi, i G {0,1,...,k} and 0o the element of order two. Then (0o, 01,0,..., 0) G Qa and (0o, 0,02,0,..., 0) G Qp. Therefore GC(G, {(0o,01,0,..., 0)},a) and GC(G, {(0o, 0,02,0,..., 0)},ft) are both generalized Cayley graphs of G. In fact, they are both isomorphic to f K2, but a and ft are not conjugate in Aut(G). Thus G is not a restricted GCI-group by Theorem 4.5. □
Theorem 5.5. Let G = Z2n, where n is an odd prime power. Then G is not a strongly GCI-group.
Proof. Let G = (a, b | an = b2 = 1, ab = ba). It can be checked that the mapping a: a ^ a-1,b ^ b is the only automorphism of G of order two. Also Qa = {aib | i G {1,..., n}} and = 0 by direct computation. Let GC(G, S, a) be any generalized Cayley graph of G. Then S C Qa. Let H = (a', b' | a'n = b'2 = 1, b'a'b' = a'-1) and ^: as ^ a's, atb ^ a'-tb. It follows that ^ is a bijection from G to H. Furthermore,
a
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GC(G,S,a) = Cay(H,y(S)). Let S = {ab,a2b}. Then y>(S) = {a-1b,a-2b}, this implies Cay(H,<p(S)) = G2n as (<p(S)> = H. Let T = {ab,a-1b}. Then (T> = G, therefore Cay(G,T) = G2n. Thus Gc(G,S,a) ^ GC(G,T, 1) = G2n, which means that G is not a strongly GCI-group from Definition 4.1.	□
Theorem 5.6. Let G = Z2n, H = D2n, where n is an odd prime power. Then G is a restricted GCI-group if H is a CI-group.
Proof. Let G = (a, b | an = b2 = 1, ab = ba> and H = (a', b' | a'n = b'2 = 1,b'a'b' = a'-1). It is easy to see that a: a ^ a-1, b ^ b is the only automorphism of G of order two. Then we have = {a®b | i e {1,..., n}} and = 0 by direct computation. Let GC(G, S, a) be any generalized Cayley graph of G. Then S C Let ^: as ^ a's,a4b ^ a'-tb. Obviously ^ is a bijection from G to H. Furthermore, GC(G, S, a) = Cay(H, y>(S)). Assume that GC(G, S1, a) ^ GC(G, S2, a), then Cay(H, ^(S1)) = Cay(H, y>(S2)). Since H is a CI-group, there exists some 7 e Aut(H) such that y(^(S1)) = y(S2). Without loss of generality, assume that there exist k, l satisfying (k, n) = 1 and 1 < l < n such that 7 is the mapping a' ^ a'k, b' ^ a'1 b'. Let ^: a ^ ak, b ^ b. Then ^ e Aut(G). Since n is some odd prime power, there must exist some 1 < m < n such that a1 = a2m. Therefore there exist ^ e Aut(G) and g = a-m such that S2 = a(g)Sj*g-1. Hence G is a restricted GCI-group.	□
Theorem 5.7. Let G be a finite cyclic group of odd order n, where n has at least two different prime divisors. Then G is not a strongly GCI-group.
Proof. Let G = G1 x G2 x • • • x Gs where Gj = Z^, p is some odd prime. Let
a: (g1, g2,..., gs) ^ ( g1, g2,..., gs). Then wa(G) = G1, and thus G \ wa(G) = 0. By Theorem 2.4, GC(G, S, a) = GC(G, S, 1), where S = G \ wa(G). It follows that G is not a GCI-group.	□
6 The GCI-groups in dihedral groups
Theorem 6.1. Let G = D2n (n > 3) be a dihedral group. If G is a restricted GCI-group, then n is some odd prime power.
Proof. Let G = D2n = (a, 6 | a" = 62 = 1,6a6-1 = a-1} be a GCI-group. Assume first that n is even. Let a: a ^ a-1, 6 ^ 6. Then a G Aut(G) is of order two. ^a(G) = {a(g-1)g | g G G} = ja® G G | i is even}.	{a(g)g = 1 | g G
G} = {a®, 6, an6 | a® G G}. It follows that Oa(G) = {a®, 6, an6 | i is odd}. This implies GC(G, {a}, a) and GC(G, {6}, a) are always generalized Cayley graphs of G. Note that they are both isomorphic to nK2. Furthermore, a(g)aYg-1 = a-2®+j if g = a® and aY = aj, a(g)aYg-1 = a-2®-j if g = a®6 and aY = aj. It follows that a(g)aYg-1 G (a}. Since G is a GCI-group, a(g)aYg-1 = 6 for some g G G and 7 G Aut(G), but this is impossible. Thus n is not even.
Assume n is odd and has at least two different prime factors, say n = p^1 • • • is the prime decomposition and t > 2. By [20, Lemma 3.4], Aut(G) = Aut(G1) x • • • x Aut(Gt), where G® = (a®, 6} and (a} = (a1} x • • • x (at}. It can be checked that there must exist two automorphisms a: a1 ^ a-1, a® ^ a®, 6 ^ 6 and ft: a2 ^ a-1, a¿ ^ a¿, 6 ^ 6 in Aut(G). Notice that each is of order two, and they are not conjugate in Aut(G) as they belong to Aut(G1) and Aut(G2) respectively, and Aut(G) is the direct product of these Aut(G®). Furthermore, 6 G üa(G) n ^(G). Thus GC(G, {6}, a) and
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GC(G, jb},ft) are two generalized Cayley graphs of G which are isomorphic to nK2. However a and ft are not conjugate. Thus n is some odd prime power.	□
Theorem 6.2. Let G = D2n, H = Z2n, with n odd prime power. Then G is a restricted GCI-group if H is a Cl-group.
Proof. Let G = (a, b | an = b2 = 1, bab = a-1) and H = Z2n = j0,1,..., 2n - 1}. We will show first that any two automorphisms of G of order two are conjugate. Let a: a ^ a®, b ^ ajb and ft: a ^ ak, b ^ alb be two automorphisms of order two. Then i, k = -1. Let y: a ^ as, b ^ a4b with (n, s) = 1. Then 7 G Aut(G). We can see that y-1 : a ^ ar, b ^ a-rtb with rs = 1 (mod n). It follows that 7a7-1: a ^ a-1, b ^ a2t+sjb. For any j, l, the equation 2t + sj = l (mod n) has a solution. It follows that a, ft are conjugate.
According to the Remark 3.2, it suffices to consider the isomorphisms of the generalized Cayley graphs induced by the same automorphisms. Without loss of generality, we consider
a: a ^ a-1, b ^ b. Let s = n—1 and I = j1,..., s}. Then (G) = ja(g-1)g | g G G} = (a). Ka = jb} U (a). Thus tta = jb} and Ua = Uie/ja®b, a-ib}.
Let GC(G, S, a) and GC(G, T, a) be any two isomorphic generalized Cayley graphs. We divide the proof into two cases.
Case 1: Qa C S.
If C S, then C T. Suppose S = Ui£/lc/ja®b,a-ib} U Qa and T = Uie/2c/ja®b, a-ib} U Let ^: G ^ H, as ^ 2s, a4b ^ n — 2t. Then ^ is a bi-jection from G to H. Furthermore, GC(G, S, a) = Cay(H, y>(S)) and GC(G, T, a) = Cay(H, ^(T)), where y(S) = Uie/lC/jn — 2i, n + 2i} U jn} and ^(T) = Uie/2c/jn — 2i, n + 2i} U jn}. Since H is a CI-group, there exists some automorphism ^ G Aut(H) such that <^(T) = ^(y(S)). Since n is the unique involution in H, ^(n) = n and ^(n — 2i) = ^(n) — 2^>(i) = n — 2^>(i) for any i G 11 C I. This implies that ^ can induce an automorphism ^ of G with rules a® ^ , b ^ b. As ^a = a^, there exist ^ and g =1 G G such that the isomorphism between GC(G, S, a) and GC(G, T, a) is a natural isomorphism.
Case 2: £ S.
If £ S, then Qa £ T. The rest of the proof is similar to that of Case 1.	□
The next result is about the graph structure. Recall that a graph r is Hamiltonian if it contains a cycle passing through all vertices of r.
Theorem 6.3. Let G = D2n with n odd prime power. Then any connected generalized Cayley graph of G is Hamiltonian.
Proof. Let H = Z2n and ^: G ^ H, as ^ 2s, a4b ^ n —2t be the bijection from G to H. Then any generalized Cayley graph GC(G, S, a) of G is isomorphic to the Cayley graph Cay(H, y(S)) of H. Therefore GC(G, S, a) is connected if and only if Cay(H, £(S)) is connected. It is well known that Cay(H, y>(S)) is connected if and only if (y(S)) = H. (y(S)) = H if and only if there exist some a®b, a-® b G S satisfying (i, n) = 1 as ^>(a®b) = n — 2i. Then there always exists a Hamilton cycle GC(G, ja®b, a-ib}, S) in a connected generalized Cayley graph of G. This completes the proof.	□
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7 Cayley regression
First of all, we give the following related definitions. Definition 7.1. Let G be a finite group.
(1)	G is called a Cayley regression if any generalized Cayley graph of G is isomorphic to some Cayley graph of G.
(2)	G is called an a-Cayley regression if any generalized Cayley graph of G induced by a € Aut(G) is isomorphic to some Cayley graph of G.
(3)	G is called a quasi-Cayley regression if any generalized Cayley graph not induced by the automorphism x ^ x-1 is isomorphic to some Cayley graph of G.
(4)	G is called an m-Cayley regression if any generalized Cayley graph of G with valency at most m is isomorphic to some Cayley graph of G.
(5)	G is called an m-quasi-Cayley regression if any generalized Cayley graph not induced by the automorphism x ^ x-1 of G with valency at most m is isomorphic to some Cayley graph of G.
(6)	G is called a skew Cayley regression if any generalized Cayley graph of G is isomorphic to some generalized Cayley graph of another finite group.
It is well known that every Cayley graph is also a generalized Cayley graph. But many examples, see [8] for instance, reveal that the converse is not true. Therefore a natural question arises.
Question 7.2. Characterize Cayley regressions.
Remark 7.3. If a: x ^ x-1 is an automorphism of G, then G is abelian. This case is very special as Ka = G and = 0. In fact, Hujdurovic et al. in [9] had already noticed this situation. They studied the relationship between the generalized Cayley graphs induced by involutory automorphism and Cayley graphs. They obtained two families of generalized Cayley graphs induced by involutory automorphisms on Z2m x Z2n and Z2 x Z2 x Z2k+1 respectively (where m > 1, n > 2, k > 1) are not vertex-transitive. Therefore we propose the definition of 'quasi-Cayley regression' and 'm-quasi-Cayley regression'. Also, we propose the following problem: Are there finite groups which are quasi-Cayley regressions but not Cayley regressions?
When G is an abelian simple group, then G is a cyclic group of prime order and Aut(G) is not necessarily a Cayley regression.
Example 7.4. For the prime p = 61, obviously, Aut(Zp) = Zp-1 = Z60. Let G = Aut(Zp), S = {±6, ±12,5, 25} and a(x) = 31x. By [17, Theorem 4.4], we have GC(G, S, a) is not a Cayley graph. Thus G is not a Cayley regression.
Theorem 7.5. Let G be a finite cyclic group of odd order n. Then G is a Cayley regression if and only if n is some odd prime power.
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Proof. The sufficiency is obvious by Theorem 5.1, it suffices to show the necessity. Assume on the contrary that n has at least two different odd prime divisors, say n = q1q2m, where q1 and q2 are different prime powers and (q1, q2) = 1, then we have G = G1 x G2 x G3, where |G11 = q1, |G2| = q2 and |G3| = m. Let a: G ^ G, (g1,g2,g3) ^ (-g1,g2,g3). It is easy to see that the order of a is two, so a can induce some generalized Cayley graphs of G. Note that wa(G) = {(g1,0,0) | g1 € G1}. Let S = {(0,1,0), (0,92 - 1,0)}. Then r = GC(G, S, a) is a generalized Cayley graph of G. Consider the vertex of the form (0,g2,g3) in r for any g2 € G2, g3 € G3. For any fixed g3, there are q2 vertices of the form {(0,g2,g3) | g2 € G2} which induce a cycle of length q2. For any other vertex of the form (g1, g2, g3) with g1 = 0, there are 2q2 vertices {(g1, g2, g3), (—g1, g2, g3) | g2 € G} which induce a cycle of length 2q2. Therefore r1 = mCq2 U (qi-1)m C2q2, which is not vertex-transitive. Thus GC(G, S, a) is not a Cayley graph, and hence G is not a Cayley regression.	□
Theorem 7.6. Let G = Zn x ■ ■ ■ x Zn with n odd, s > 2. Then G is not a Cayley
regression.
Proof. Let a: (i1, i2,i3 ..., is) ^ (i2, i1, i3,..., is) for all it € Zn. So a € Aut(G) and o(a) =2. Therefore a can induce generalized Cayley graphs of G. Let S = {(1, 0,..., 0), (0, n - 1,0,..., 0)}. It follows that GC(G, S, a) is a generalized Cayley graph of G.
Consider vertex (0,..., 0), then vertices like (i, i, 0,..., 0) and (i, i - 1,0,..., 0) are in the same cycle with (0,..., 0). Thus (0,..., 0) is in a cycle of length 2n.
Consider vertex (0, n-1, 0,..., 0), then vertices like (i, n-1 + i, 0,..., 0) and (^ + i, i,..., 0) are in the same cycle with (0, n-1, 0,..., 0). It follows that (0, n-1, 0,..., 0) is in a cycle of length n.
Therefore GC(G, S, a) is not vertex-transitive, and thus GC(G, S, a) is not a Cayley graph. That completes the proof.	□
From Theorem 7.5, we see that the cyclic group of odd non prime power order is not an m-Cayley regression for any m > 0. So we only consider the cyclic groups of even order in the rest of this section.
Corollary 7.7 ([9]). Let G = Z2n. Then GC(G, S, a) is isomorphic to a Cayley graph on D2n, where a: x ^ -x.
According to Corollary 7.7, we can see that Z2pn (with p an odd prime) is a skew-Cayley regression since a: x ^ -x is the only automorphism of G of order two.
Theorem 7.8. Let G be a finite cyclic group of order 2n with n > 3. Then
(1)	G is a 3-quasi-Cayley regression;
(2)	G is a 4-quasi-Cayley regression if and only if n = 3.
Proof. Assume that G = {0,1,..., 2n - 1}. By Theorem 5.2, we have that a: x ^ (2n-1 - 1)x and P: x ^ (2n-1 + 1)x are the only two automorphisms of G of order two except the automorphism x ^ -x. Also, the valency of the generalized Cayley of G induced by a or P are even as x + a(x) = 0 and x + P(x) =0 for any x € G.
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(1) Consider the generalized Cayley graphs induced by a, ^ respectively. For any g e G,
wa(G) = {a(-g)g | g e G} = {(2n-1 - 1)(-g)+ g | g e G}
= {2n-1g + 2g (mod 2") | g e G} = {g | g = 0 (mod 2)} = Ka.
wg(G) = {£(-g)g | g e G} = {(2n-1 + 1)(-g)+ g (mod 2") | g e G} = {-2"_1g (mod 2") | g e G} = {0, 2n-1} = Kg.
It follows that for any generalized Cayley graph GC(G, S, a), S contains no even integers and, for any generalized Cayley graph GC(G, S, ,0), 0 and 2"_1 are not contained in S. For any g e G with g odd,
a(-g) =	(2n-1 - 1)(-g) (mod 2")
=	g - 2n-1g (mod 2")
=	2"g + g - 2"_1g (mod 2")
=	2"_1g + g (mod 2")
=	2"_1 + g (mod 2").
This implies that there are 2"_2 couples can be included in S, that is, S1 — {1,2" 1 + 1},
S3 = {3, 2n-1 + 3},..., S2n-i_1 = {2n-1 - 1, 2" - 1}. For any g e G \ wg(G),
£(-g) = (2n-1 + 1)(-g) (mod 2") = -g - 2"_1g (mod 2") = 2"g - g - 2"_1g (mod 2") = 2n-1g - g (mod 2").
Then we have
12" - g, if g is even.
This implies that there are (2n-1 - 1) couples which could be included in S, they are:
S1 = {1, 2"_1 - 1}, S3 = {3, 2"_1 - 3},
S2n-2_1 = {2"_2 - 1, 2"_2 + 1}, S2n-i+1 = {2"_1 + 1, 2" - 1},
S2n-1+2n-2_1 = {2"_1 + 2"_2 - 1, 2"_1 + 2"_2 + 1}, T2 = {2, 2" - 2},
T2n-1_2 = {2"_1 - 2, 2"_1 + 2}.
Let r = GC(G, S, a), where S = S,, then r = 2"_2C4 by Theorem 5.2. Let r = GC(G,S,£). If S = Si, we have GC(G,S,£) ^ C2~ as S, is the left generating set for (G, *) by Lemma 2.11. If S = T,, then GC(G, S, £) is isomorphic to 2"_fcC2fc
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if 2ki = 0 (mod 2n); and isomorphic to C2n otherwise. In conclude, all of the 2-valent generalized Cayley graphs of G induced by a or ft are Cayley graphs and, this implies that G is a 3-quasi-Cayley regression.
(2) When n = 3, it is easy to check that those 4-valent generalized Cayley graphs induced by a and respectively, are all Cayley graphs. Next we construct a family of generalized Cayley graphs which is not vertex-transitive to show the necessity.
Let S = Si U Tj, where i € {1,..., 2n-2 - 1} U {2n-1 + 1,..., 2n-1 + 2n-2 - 1} is odd and j € {2,..., 2n-1 - 2} is even. If x is odd, then N(x) = {2n-1 + x + i, x -i, 2n-1 + x + j, 2n-1 + x - j }. If x is even, then N (x) = {x + i, 2n-1 + x - i, x+j, x - j }. Suppose X is the bicirculant such that the vertex set V(X) can be partitioned into to subsets U = {uk | k € Z2n-i} and V = {vk | k € Z2n-i}, and there is an automorphism of X such that p(uk) = wk+1 and p(vk) = vk+1, k € Z2n-i. The edge set E(X) can be partitioned into three subsets:
L = Ukez2„-i {uk, uk+i | l € L}, M = Ukez2„-i {uk, vk+m | m € M}, R = Ukez2„-i {vk, vk+r | r € R},
so we have L = {±(2n-2 + |)}, M = {2n-2 + i+1, -^}, R = {±§}. Then X = BC2n-i [L, M, R]. Let 7 be the mapping as follows:
{x ^ u x-i, if x is odd; x ^ v x, if x is even.
It follows that r = X. Note that BC2n-i [L, M, R] = BC2,-i [aL, aM + b, aR] with a, 6 € Z2n-i and a invertible [12]. Then r = BC2n-i [L, M', R] with M' = M + ^ = {0, 2n-2 + i}. In particular, r is connected since (L, M', R) = Z2n-i. When j = 2i, there are no triangles with three vertices of the form {uk, Vk+2n-2+
¿+i, vk- i-i }, but there is a triangle with three vertices as {vk/, uk,+ i-i, uk,_2„-2_i+i } since for n > 3,
k + 2n-2 + 1+1 ± j ^ k - i-1 (mod 2n-1) 2 2 ^ 2 v 7
k' + - (V-2 + 0 = k' - 2n-2 -	(mod 2n-1).
This implies that there is no automorphism of X which permutates uk and vk/. So X is not vertex-transitive when n > 3. This completes the proof.	□
At last, we propose the following questions for further research.
Question 7.9. Classify finite GCI-groups, such as Zm where m is odd with at least two different prime divisors, abelian groups, dihedral groups and some classes of finite simple groups.
Question 7.10. Characterize the structure of the automorphism group of any generalized Cayley graph.
Question 7.11. Classify Cayley regressions for certain types of group, such as the cyclic groups and the dihedral groups.
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