1-REGULAR CAYLEY GRAPHS of VALENCY 7 06 07 JING JIAN LI ˛, GENG RONG ZHANG and BO LING 08

Bull. Aust. Math. Soc. 0 (2013), 1–7 doi:10.1017/S0004972713000087 01

04 05 1-REGULAR CAYLEY GRAPHS OF VALENCY 7 06 07 JING JIAN LI ˛, GENG RONG ZHANG and BO LING 08

09 (Received 15 November 2012; accepted 27 December 2012) 10

11 Abstract 12

13 A graph Γ is called 1-regular if AutΓ acts regularly on its arcs. In this paper, a classiﬁcation of 1-regular Cayley graphs of valency 7 is given; in particular, it is proved that there is only one core-free graph up to 14 isomorphism. 15

16 2010 Mathematics subject classiﬁcation: primary 05C25.

17 Keywords and phrases: 1-regular, Cayley graph, core-free, small valency.

20 1. Introduction Q1 21 Throughout this paper, all graphs are ﬁnite, simple and undirected. 22 Let Γ be a graph. We denote the vertex set, edge set, arc set and full automorphism 23 group by V(Γ), E(Γ), Arc(Γ) and AutΓ, respectively. Γ is called X-vertex-transitive, 24 X-edge-transitive and X-arc-transitive if X acts transitively on V(Γ), E(Γ), and Arc(Γ) 25 respectively, where X ≤ AutΓ. We simply call Γ vertex-transitive, edge-transitive and 26 arc-transitive for the case where X = AutΓ. In particular, Γ is called (X, 1)-regular if 27 X ≤ AutΓ acts regularly on its arcs, and 1-regular when X = AutΓ. Let G be a ﬁnite 28 group with identity 1. Γ is called a Cayley graph of G, denoted by Γ = Cay(G, S ), − − 29 if there is a subset S of G with 1 < S and S = S 1:={s 1 | s ∈ S } such that V(Γ) = G 30 and E(Γ) = {(g, sg) | g ∈PROOFG, s ∈ S }. It is easy to see that Cay(G, S ) has valency |S |. 31 Moreover, Cay(G, S ) is connected if and only if hS i = G. 32 For a Cayley graph Γ = Cay(G, S ), G can be viewed as a regular subgroup of AutΓ 33 by right multiplication action on V(Γ). For convenience, we still denote this regular 34 subgroup by G. Then a Cayley graph is vertex-transitive. If G is a normal subgroup

35 of AutΓ, then Γ is called a normal Cayley graph of G. While for a nonnormal Cayley

36 graph Γ, if AutΓ contains a normal subgroup N that is semi-regularly on V(Γ) and

37 has exactly two orbits, then Γ is called a bi-normal Cayley graph. Both normal

38 and bi-normal Cayley graphs have nice properties (see, for example, [5,6,7, 11]).

39 The project was sponsored by the Scientiﬁc Research Foundation of Guangxi University (Grant No. 40 XBZ110328), NSF of Guangxi (Nos 2012GXNSFBA053010 and 2010GXNSFA013109), NSF of China

41 (No. 10961007). c 42 2013 Australian Mathematical Publishing Association Inc. 0004-9727/2013 $16.00

Marked Proof Ref: 55654 February 5, 2013 2 J. J. Li, G. R. Zhang and B. Ling [2]

01 And Cay(G, S ) is called core-free (with respect to G) if G is core-free in some T x 02 X ≤ Aut(Cay(G, S )), that is, CoreX(G):= x∈X G = 1. 03 Li proved in [6] that there are only a ﬁnite number of core-free s-transitive Cayley

04 graphs of given valency k for s ∈ {2, 3, 4, 5, 7} and k ≥ 3, and that, with the exceptions

05 s = 2 and (s, k) = (3, 7), every s-transitive Cayley graph is a normal cover of a core-free

06 one. Li and Lu gave a classiﬁcation of cubic s-transitive Cayley graphs for s ≥ 2 in [8].

07 What about the case where s = 1? Until now, the results on 1-regular graphs have

08 mainly focused on constructing examples. For example, Frucht gave the ﬁrst example

09 of cubic 1-regular graphs in [4]. Conder and Praeger then constructed two inﬁnite

10 families of cubic 1-regular graphs in [2]. Marusi˘ c˘ and Malnic˘ in [9, 10] constructed

11 two inﬁnite families of tetravalent 1-regular graphs. Classiﬁcation of such graphs has received great interest in recent years. Motivated by the above results and problem, we 12 consider 1-regular Cayley graphs in this paper. We present the following theorem. 13

14 T 1.1. Let Γ = Cay(G, S ) be a 1-regular Cayley graph with valency 7, and let 15 N = CoreA(G). Then Γ is connected, and one of the following holds: 16 (1) G = N and Γ is a normal Cayley graph; 17 (2) |G : N| = 2, Γ is a bi-normal Cayley graph; 18 (3) Γ is a normal cover of a core-free graph (up to isomorphism), (A, G) = (S7, S6). 19 R 1.2. For more information on the core-free graph in (3) of Theorem 1.1, the 20 readers can refer to Lemmas 2.2 and 2.3. 21

23 2. Core-free case 24 In this section, we will consider the core-free 1-regular Cayley graphs of valency 7. 25 First, we will list all the (X, 1)-regular graphs with automorphism group X containing 26 the regular subgroup. For an (X, 1)-regular graph, one does not expect it also to be 1- 27 regular. So it is an important task for us to determine whether an (X, 1)-regular graph 28 is 1-regular. 29 Let X be an arbitrary ﬁnite group with a core-free subgroup H. For an element 2 30 g ∈ X \ H such that g ∈ H, thePROOF coset graph Cos(X, H, g) is the graph with vertex set −1 31 [X : H], and two vertices Hx, Hy are adjacent if and only if y x ∈ HgH. By [8], we 32 have the following simple proposition.

33 P 2.1. Let Γ = Cay(G, S ) be a connected (X, 1)-regular Cayley graph of 34 valency 7 with G ≤ X ≤ AutΓ. Let H be the stabiliser of 1 ∈ V(Γ) in X. Then there 35 exists an involution τ in S such that τ ∈ X \ NX(H), Γ Cos(X, H, τ), hτ, Hi = X, 36 S = G ∩ HτH and G = hS i. 37

38 Let Γ = Cay(G, S ) be a connected (X, 1)-regular core-free Cayley graph of valency

39 7, where G ≤ X ≤ AutΓ. For convenience, we let Σ = {1, 2,..., 7}. By considering

40 the right multiplication action of X on the right cosets of G in X, we may view X

41 as a subgroup of the symmetric group S7, which contains a regular subgroup (of S7)

42 isomorphic to a stabiliser of X acting on V(Γ). And in this way, G is a stabiliser of X

Marked Proof Ref: 55654 February 5, 2013 [3] 1-regular Cayley graphs 3

01 acting on Σ by marking [X : G] as Σ. Without loss of generality, we may assume that

02 G ﬁxes 1.

03 In the following, we will construct all possible connected core-free (X, 1)-regular

04 Cayley graphs of valency 7 with a given stabiliser H Z7. Without loss of generality

05 we let H = hσi with σ = (1 2 3 4 5 6 7). Clearly H acts regularly on Σ; then H is a

06 regular subgroup of S7. By Proposition 2.1, we may take an involution τ ∈ S7 \ NS7 (H) τ σ σ 07 such that 1 = 1. Let X = hτ, Hi, S = {σ ∈ HτH | 1 = 1}; then G = hS i = {σ ∈ X | 1 =

08 1} is a complement subgroup of H in X. Thus G acts regularly on [X : H], and it follows

09 that Cos(X, H, τ) Cay(G, S ) is a connected core-free (X, 1)-regular Cayley graph of G 10 . Note that all isomorphic regular subgroups of S7 are conjugate in S7 (see, for example, [12]), and Cos(X, H, τ) is independent of the choice of H up to isomorphism. 11 It is well known that Cos(X, H, τ) Cos(Xσ, H, τσ) for any σ ∈ N (H). With the 12 S7 help of GAP, we can get that there are in total 74 such τ0 s, which are conjugate under 13 NS (H) to one of the following nine permutations: 14 7

15 τ7,1 = (2 7), τ7,2 = (2 4)(3 7), τ7,3 = (2 5)(3 7), 16 τ7,4 = (2 6)(3 7), τ7,5 = (2 3)(5 7), τ7,6 = (2 6)(3 7)(4 5), 17 τ7,7 = (2 6)(3 4)(5 7), τ7,8 = (2 3)(4 6)(5 7), τ7,9 = (2 7)(3 5)(4 6). 18 σ 19 For ı ∈ {1, 2,..., 9}, we let Γ7,ı = Cos(X7,ı, H, τ7,ı) and G7,ı = {σ ∈ X7,ı | 1 = 1}, where 20 X7,ı = hτ7,ı, σi. Then Γ7,ı Cay(G7,ı, S 7,ı) with S 7,ı = G7,ı ∩ Hτ7,ıH and G7,ı = hS 7,ıi. 21 L 2.2.( G7,2, X7,2) (S4, PSL(3, 2)), (G7,i, X7,i) (A6, A7) and (G7, j, X7, j) 22 (S6, S7), where i ∈ {3, 4, 5} and j ∈ {1, 6, 7, 8, 9}. 23

24 P. Let i ∈ {3, 4, 5}, j ∈ {1, 6, 7, 8, 9} and

25 −1 4 π1 = (τ7,1σ ) τ7,1, β1 = τ7,1, 26 3 2 3 −3 2 −3 2 2 27 π6 = (τ7,6σ ) (τ7,6σ) σ , β6 = τ7,6σ (τ7,6σ ) τ7,6σ τ7,6, 3 −2 −3 2 28 π7 = τ7,7σ τ7,7σ τ7,7, β7 = στ7,7σ (τ7,7σ) , 29 −1 2 2 −2 −2 2 −1 2 π8 = (σ τ7,8) σ(στ7,8) σ τ7,8, β8 = (τ7,8σ ) τ7,8σ τ7,8σ τ7,8σ, 30 −1 4 PROOF2 −2 −1 2 2 π = (τ σ ) τ , β = σ τ σ τ σ τ σ (τ σ) . 31 9 7,9 7,9 9 7,9 7,9 7,9 7,9

32 Then π j = (2 3 4 5 6 7) and β j = (2 7). Note that π j acts transitively on Σ \{1}, and X7, j 33 acts 2-transitively on Σ. On the other hand, X7, j contains a 2-cycle (2 7), which leads 34 to X7, j S7 by [3, Theorem 3.3A]. Furthermore, G7, j S6. 35 Let

36 π = σ−1τ σ−2, β = σ2(τ σ)2, 37 3 7,3 3 7,3 −1 −2 2 2 38 π4 = σ τ7,4σ , β4 = (τ7,4σ ) ,

39 −1 3 −1 2 π5 = σ τ7,5σ , β5 = (στ7,5σ τ7,5) . 40

41 Then π3 = (2 6 7 4 5), π4 = (2 6 3 4 5), π5 = (2 4 5 7)(3 6) and βi = (1 2 3). Noticing that

42 hτ7,i, πii acts transitively on Σ \{1}, X7,i acts 2-transitively on Σ. However, X7,i contains

Marked Proof Ref: 55654 February 5, 2013 4 J. J. Li, G. R. Zhang and B. Ling [4]

01 a 3-cycle (1 2 3) and all generators of X7,i are even permutations, thus X7,i A7 by [3, 02 Theorem 3.3A], and, moreover, G7,i A6. 2 2 −2 2 03 Let µ = (σ τ7,2) = (1 4)(6 7); then τ7,2 = µσ µσ µ and X7,2 = hτ7,2, σi = hµ, σi. 7 4 4 3 2 04 Note that σ = (σ µ) = (σµ) = µ = 1, X7,2 PSL(3, 2) and G7,2 S4 by [1].

06 In the rest of this section, we let

08 c1=(1 2 3 4 5 6 7)(8 9 10 11 12 13 14),

09 c2=(1 8)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9),

10 d1=(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21), 11 d2=(1 8 15)(2 10 19)(3 12 16)(4 14 20)(5 9 17)(6 11 21)(7 13 18), 12 τ¯7,1=(3 5)(11 13), τ¯7,2=(3 4)(5 7)(9 11)(12 13), 13

14 τ¯7,3=(3 4)(6 7)(9 10)(12 13), τ¯7,4=(4 5)(6 7)(9 10)(11 12),

15 τ¯7,5=(2 7)(4 5)(9 14)(11 12), τ¯7,6=(2 7)(3 4)(5 6)(9 14)(10 11)(12 13),

16 τ¯7,7=(2 6)(3 4)(5 7)(9 13)(10 11)(12 14)(16 20)(17 18)(19 21), 17 τ¯7,9=(2 7)(3 5)(4 6)(9 14)(10 12)(11 13). 18

19 Let H¯1 = hc1, c2i, X¯7,ı = hc1, c2, τ¯7,ıi, Γ¯ 7,ı = Cos(X¯7,ı, H¯1, τ¯7,ıi, S¯ 7,ı = {σ ∈ H¯1τ¯7,ıH¯1 | 20 σ ¯ ¯ 7 2 c2 1 = 1} and G7,ı = hS 7,ıi, with ı = 1, 2, 3, 4, 5, 6, 9. Since c1 = c2 = 1 and c1 = 21 −1 ¯ ¯ ¯ ¯ ¯ c1 , H1 D14. Write Σ1 = {1, 2,..., 14}, Σ2 = {1, 2,..., 21}, H2 = hd1, d2i, X7,7 = σ 22 hd1, d2, τ¯7,7i, Γ¯ 7,7 = Cos(X¯7,7, H¯2, τ¯7,7i, S¯ 7,7 = {σ ∈ H¯2τ¯7,7H¯2 | 1 = 1} and G¯7,7 = d 23 ¯ 7 3 2 2 ¯ hS 7,7i. Since d1 = d2 = 1 and d1 = d1, H2 Z7 o Z3. 24

25 L 2.3. If ı ∈ {1, 2, 3, 4, 5, 6, 7, 9}, then Γ7,ı is not a core-free 7-valent 1-regular

26 Cayley graph.

27 P. For convenience, we denote S¯ ,ı = {s¯ | s ∈ S ,ı} with ı ∈ {1, 2, 3, 4, 5, 6, 7, 9}. 28 7 7 According to our calculations, 29

30 −1 PROOF2 −2 3 3 −2 2 −1 S 7,1={a7,1, b7,1, b , b a7,1b , b a7,1b , b a7,1b , b7,1a7,1b }, 31 7,1 7,1 7,1 7,1 7,1 7,1 7,1 7,1 − − − 32 1 1 1 S 7,2={τ7,2, a7,2, b7,2, a7,2, b7,2, τ7,2b7,2, a7,2b7,2}, 33 S ={τ , a , b , a−1 , b−1 , τ a b τ a−1 , τ a2 b−2 }, 34 7,3 7,3 7,3 7,3 7,3 7,3 7,3 7,3 7,3 7,3 7,3 7,3 7,3 7,3

35 −1 −1 −1 −1 −1 S 7,4={a7,4, b7,4, c7,4, b7,4, c7,4, b7,4a7,4b7,4, c7,4b7,4a7,4b7,4c7,4}, 36 { −1 −1 −1 −1 −1 −1 } 37 S 7,5= a7,5, b7,5, c7,5, b7,5, c7,5, b7,5c7,5a7,5c7,5b7,5, b7,5c7,5a7,5c7,5b7,5 ,

38 −1 −1 2 −1 −1 −2 S 7,6={τ7,6, a7,6, b7,6, a7,6, b7,6, a7,6b7,6a7,6b7,6, τ7,6b7,6a7,6b7,6}, 39 −1 −1 −1 −1 40 S 7,7={τ7,7, a7,7, b7,7, a7,7, b7,7, τ7,7b7,7τ7,7a7,7τ7,7, τ7,7a7,7τ7,7b7,7τ7,7}, 41 −1 −1 −1 −1 S 7,9={τ7,9, a7,9, b7,9, a7,9, b7,9, a7,9b7,9τ7,9, τ7,9b7,9a7,9}, 42

Marked Proof Ref: 55654 February 5, 2013 [5] 1-regular Cayley graphs 5

01 where a7,1 = (3 5),a ¯7,1 = τ¯7,1, a7,4 = (3 6)(4 7),a ¯7,4 = τ¯7,4, a7,5 = (2 7)(4 5),a ¯7,5 = τ¯7,5 02 and

03 b7,1 = (2 7 5 3 6 4), b¯7,1 = (2 7 5 3 6 4)(8 13 11 14 12 10), 04

05 a7,2 = (2 6 3)(4 5 7), a¯7,2 = (2 7 4)(3 6 5)(8 14 10)(9 13 11),

06 b7,2 = (2 4)(3 5 7 6), b¯7,2 = (2 7 6 5)(3 4)(8 14)(9 13 12 11), 07

08 a7,3 = (2 6 7 4 5), a¯7,3 = (2 7 6 5 3)(8 13 12 11 9),

09 b7,3 = (2 5 4 7 3), b¯7,3 = (2 3 4 6 7)(8 9 10 12 13), 10 b = (2 6 3 4 5), b¯ = (2 3 4 5 7)(8 9 10 12 14), 11 7,4 7,4

12 c7,4 = (3 7 4 5 6), c¯7,4 = (2 4 5 6 7)(9 10 11 12 14), 13 b7,5 = (2 4 5 7)(3 6), b¯7,5 = (2 7 5 4)(3 6)(9 14 12 11)(10 13), 14

15 c7,5 = (2 3 6 7)(4 5), c¯7,5 = (2 7 6 3)(4 5)(9 14 13 10)(11 12),

16 a7,6 = (2 6)(3 4 5), a¯7,6 = (2 7 4)(5 6)(9 14 11)(12 13), 17 ¯ 18 b7,6 = (2 6 4 5 3 7), b7,6 = (2 7 4 3 6 5)(9 14 11 10 13 12),

19 a7,9 = (2 7 3 6), a¯7,9 = (2 6 3 7)(9 13 10 14), 20 b = (2 6 3 5), b¯ = (3 6 4 7)(10 13 11 14), 21 7,9 7,9

22 a7,7 = (2 4 7 5 6 3), b7,7 = (2 4 5 3 7 6), 23 a¯7,7 = (2 5 6 4 3 7)(9 12 13 11 10 14)(16 19 20 18 17 21), 24

25 b¯7,7 = (2 4 7 5 6 3)(9 11 14 12 13 10)(16 18 21 19 20 17).

26 Note thatτ ¯7,ı ∈ S¯ 7,ı,τ ¯7,ı ∈ G¯7,ı, and it follows that X¯7,ı = G¯7,ıH¯1. It is easy to see 27 G¯7,ı that H¯1 acts regularly on Σ¯ 1 and 1 = 1. It follows that G¯7,ı ∩ H¯1 = 1 and G¯7,ı 28 is a complement subgroup of H¯1 in X¯7,ı. Hence, we have Γ¯ 7,ı Cay(G¯7,ı, S¯ 7,ı) 29 for ı ∈ {1, 2, 3, 4, 5, 6, 9}. With a similar argument, we get Γ¯ 7,7 Cay(G¯7,7, S¯ 7,7). 30 −→ PROOF−→ ¯ −→ −→ −→ ¯ Let Φ7,1 : a7,1 a¯7,1, b7,1 b7,1; Φ7,i : τ7,i τ¯7,i, a7,i a¯7,i, b7,i b7,i; Φ7, j : 31 a7, j −→ a¯7, j, b7, j −→ b¯7, j, c7, j −→ c¯7, j with i ∈ {2, 3, 6, 7, 9} and j ∈ {4, 5}. 32 According to the proof of the above paragraph, ha7,1, b7,1i S6, then ha¯7,1, b¯7,1i is 33 also isomorphic to S6 by replacing a7,1 and b7,1 witha ¯7,1 and b¯7,1, respectively. Thus 34 ¯ ¯ Φ7,1 ¯ G7,1 G7,1, that is, Φ7,1 is an isomorphism of G7,1 to G7,1 denoted by G7,1 = G7,1. 35 Φ7,1 Φ7,ı Φ7,ı Furthermore, S = S¯ 7,1. With similar arguments, we have G = G¯7,ı and S = 36 7,1 7,ı 7,ı S¯ . Then Cay(G , S ) Cay(G¯ , S¯ ), that is, Γ Γ¯ . Since H¯ D and 37 7,ı 7,ı 7,ı 7,ı 7,ı 7,ı 7,ı 1 14 H¯2 Z7 o Z3, Γ¯ 7,ı is not 1-regular and so Γ7,ı for ı ∈ {1, 2, 3, 4, 5, 6, 7, 9}. 38 39 3. The proof of Theorem 1.1 40

41 In this section, we will prove our main result. First, we need some deﬁnitions and

42 properties, which will be used in the following.

Marked Proof Ref: 55654 February 5, 2013 6 J. J. Li, G. R. Zhang and B. Ling [6]

01 Assume that Γ is an X-vertex-transitive graph. Let N be a normal subgroup of X. 02 Denote the set of N-orbits in V(Γ) by VN. The normal quotient ΓN of Γ induced by N 03 is deﬁned as the graph with vertex set VN, and two vertices B, C ∈ VN are adjacent if

04 there exist u ∈ B and v ∈ C such that they are adjacent in Γ. It is easy to show that X/N

05 acts transitively on the vertex set of ΓN. Assume further that Γ is X-edge-transitive.

06 Then X/N acts transitively on the edge set of ΓN, and the valency val(Γ) = mval(ΓN)

07 for some positive integer m. If m = 1, then Γ is called a normal cover of ΓN.

08 We are now in a position to prove Theorem 1.1. Let Γ = Cay(G, S ) be a 1- 09 regular Cayley graph of valency 7. Then it is trivial to see that Γ is connected. Let

10 A = AutΓ and N = CoreX(G) be the core of G in X. Assume that N is not trivial.

11 Then either G = N or |G : N| ≥ 2. The former implies G E X, that is, Γ is a normal

12 Cayley graph respect to G. For the case where |G : N| = 2, it is easy to see that Γ

13 is a bi-normal Cayley graph. Suppose that |G : N| > 2, namely, N has at least three

14 orbits on V(Γ). Consider the normal quotient ΓN; we have that ΓN is a Cayley graph

15 of G/N, G/N ≤ X/N . AutΓN and ΓN is core-free with respect to G/N. Clearly ΓN

16 is an X/N-arc-transitive Cayley graph of G/N if Γ is X-arc-transitive, and under this

17 assumption, Γ is a normal cover of ΓN. Now suppose that N is trivial; then Γ is

18 core-free. According to Lemmas 2.2 and 2.3, there is only one possible core-free 1- regular Cayley graph of valency 7 (up to isomorphism): (G, X) (S , S ). The proof 19 6 7 of Theorem 1.1 is complete. 20

24 References

26 [1] J. H. Conway, R. T. Curtis, S. P. Noton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups (Clarendon Press, Oxford, 1985). 27 [2] M. D. E. Conder and C. E. Praeger, ‘Remarks on path-transitivity in ﬁnite graphs’, European J. 28 Combin. 17 (1996), 371–378. 29 [3] J. D. Dixon and B. Mortimer, Permutation Groups, 2nd edition (Springer-Verlag, New York, 30 1996). PROOF [4] R. Frucht, ‘A one-regular graph of degree three’, Canad. J. Math. 4 (1952), 240–247. 31 [5] C. H. Li, ‘Finite s-arc transitive graphs of prime-power order’, Bull. Lond. Math. Soc. 33 (2001), 32 129–137. 33 [6] C. H. Li, ‘Finite s-arc transitive Cayley graphs and ﬂag-transitive projective planes’, Proc. Amer.

34 Math. Soc. 133 (2005), 31–40. [7] C. H. Li, ‘Finite edge-transitive Cayley graphs and rotary Cayley maps’, Trans. Amer. Math. Soc. 35 359 (2006), 4605–4635. 36 [8] J. J. Li and Z. P. Lu, ‘Cubic s-transitive Cayley graphs’, Discrete Math. 309 (2009), 6014–6025. 37 [9] D. Marusi˘ c,˘ ‘A family of one-regular graphs of valency 4’, European J. Combin. 18 (1997), 59–64.

38 [10] A. Malnic,˘ D. Marusi˘ c˘ and N. Seifter, ‘Constructing inﬁnite one-regular graphs’, European J. Combin. 20 (1999), 845–853. 39 [11] C. E. Praeger, ‘Finite normal edge-transitive Cayley graphs’, Bull. Aust. Math. Soc. 60 (1999), 40 207–220. 41 [12] S. J. Xu, X. G. Fang et al., ‘5-arc transitive cubic Cayley graphs on ﬁnite simple groups’, European

42 J. Combin. 28 (2007), 1023–1036.Q2

Marked Proof Ref: 55654 February 5, 2013 [7] 1-regular Cayley graphs 7

01 JING JIAN LI, School of Mathematics and Statistics, 02 Yunnan University, Kunming, Yunnan 650031, PR China 03 and 04 College of Mathematics and

05 Information Science, Guangxi University,

06 Nanning 530004, PR China

07 e-mail: [email protected]

09 GENG RONG ZHANG, College of Mathematics and

10 Information Science, Guangxi University,

11 Nanning 530004, PR China

12 e-mail: [email protected]

13 BO LING, College of Mathematics and 14 Information Science, Guangxi University, 15 Nanning 530004, PR China 16 e-mail: [email protected] 17

29 30 PROOF 31

Marked Proof Ref: 55654 February 5, 2013 01 AUTHOR QUERIES

03 04 Q1 (page 1) 05 Please check corresponding author given is ok as set. 06 07 Q2 (page 6) 08 How many authors for refrence [12]? Please suply a list of all the authors unless there 09 are more than 10 in which case we will only need the ﬁrst author followed by et al. 10

29 30 PROOF 31

Marked Proof Ref: 55654 February 5, 2013