- Email: [email protected]
The automorphism group of the alternating group graph
Applied Mathematics Letters 24 (2011) 229–231
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
The automorphism group of the alternating group graph Jin-Xin Zhou Department of Mathematics, Beijing Jiaotong University, Beijing 100044, PR China
article
info
Article history: Received 2 May 2009 Received in revised form 14 September 2010 Accepted 15 September 2010 Keywords: Alternating group graph Automorphism group Cayley graph
abstract Let An be the alternating group of degree n with n ≥ 3. Set S = {(1 2 i), (1 i 2) | 3 ≤ i ≤ n}. The alternating group graph, denoted by AGn , is defined as the Cayley graph on An with respect to S. Jwo et al. (1993) [J.-S. Jwo, S. Lakshmivarahan, S.K. Dhall, A new class of interconnection networks based on the alternating group, Networks 23 (1993) 315–326] introduced the alternating group graph AGn as an interconnection network topology for computing systems, and they proved that AGn is arc-transitive. In this work, it is shown that the full automorphism group of AGn is the semi-direct product R(An ) o Aut(An , S ), where R(An ) is the right regular representation of An and Aut(An , S ) = {α ∈ Aut(An ) | S α = S } ∼ = Sn−2 × S2 . It follows from this result that AGn is arc-transitive but not 2-arc-transitive. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Graphs considered in the work are finite, connected, simple and undirected. For a graph X , denote by Aut(X ) the full automorphism group of X . An s-arc in a graph X is an ordered (s + 1)-tuple (v0 , v1 , . . . , vs−1 , vs ) of vertices of X such that vi−1 is adjacent to vi for 1 ≤ i ≤ s and vi−1 ̸= vi+1 for 1 ≤ i ≤ s − 1. A graph X is said to be s-arc-transitive if Aut(X ) is transitive on the set of s-arcs in X . In particular, 0-arc-transitivity means vertex-transitivity, and 1-arc-transitivity means arc-transitivity. For a set V and a group G with identity element 1, an action of G on V is a mapping V × G → V , (v, g ) → v g , such that v 1 = v and (v g )h = v gh for v ∈ V and g , h ∈ G. The subgroup K = {g ∈ G | v g = v, ∀v ∈ V } of G is called the kernel of G acting on V . For two groups K , H, if H acts on K (as a set) such that (xy)h = xh yh for any x, y ∈ K and h ∈ H, then H is said to act on K as a group. In this case, we use K o H to denote the semi-direct product of K with H with respect to the action. For a finite group G and a subset S of G such that 1 ̸∈ S and S = S −1 (where 1 is the identity element of G), the Cayley graph Cay(G, S ) on G with respect to S is defined to have vertex set G and edge set {{g , sg } | g ∈ G, s ∈ S }. The automorphism group Aut(Cay(G, S )) of Cay(G, S ) contains the right regular representation R(G) of G, the action of G on itself by right multiplication, as a subgroup. Thus Cay(G, S ) is vertex-transitive. Furthermore, Aut(G, S ) = {α ∈ Aut(G) | S α = S } is a subgroup of Aut(Cay(G, S ))1 , the stabilizer of the vertex 1 in Aut(Cay(G, S )). A Cayley graph Cay(G, S ) is said to be normal if R(G) is normal in Aut(Cay(G, S )). Xu [1, Proposition 1.5] proved that Cay(G, S ) is normal if and only if Aut(Cay(G, S ))1 = Aut(G, S ). Let An be the alternating group of degree n with n ≥ 3. Set S = {(1 2 i), (1 i 2) | 3 ≤ i ≤ n}. The alternating group graph, denoted by AGn , is defined as the Cayley graph AGn = Cay(An , S ).
(1)
Jwo et al. [2] introduced the alternating group graph as an interconnection network topology for computing systems. Following this pioneering article, the alternating group graph has been extensively studied over decades by many authors. For example, Chang and Yang [3] investigated the panconnectivity, and Hamiltonian-connectivity in alternating group
E-mail address: [email protected]. 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.09.009
230
J.-X. Zhou / Applied Mathematics Letters 24 (2011) 229–231
graphs, and they [4] also studied the fault-tolerant Hamiltonicity of alternating group graphs. Teng et al. [5] proved that if n ≥ 3 then for any two different vertices x and y of AGn and for any integer ℓ satisfying d(x, y) ≤ ℓ ≤ |AGn | − d(x, y), there exists a Hamiltonian cycle C of AGn such that the relative distance between x, y on C is ℓ, where d(x, y) is the distance between x and y in AGn . It is known from [2] that AGn is arc-transitive. In many situations, it is highly advantageous to use interconnection networks which are highly symmetric. This often simplifies the computational and routing algorithms. When dealing with the symmetry of graphs, the goal is to gain as much information as possible about structure properties of full automorphism groups of graphs. Recently, several publications have been focused on investigation of automorphism groups of Cayley graphs having connection with interconnection networks. For example, let Sn be the symmetric group of degree n, and set T1 = {(i i + 1)|1 ≤ i ≤ n − 1}, T2 = T1 ∪ {(1 n)} and T3 = {(1 i)|2 ≤ i ≤ n}. The Cayley graphs Cay(Sn , Tj )(1 ≤ j ≤ 3) are called the bubble-sort graph, modified bubble-sort graph and star graph, respectively, in [6]. Huang and Zhang [7,8] determined the full automorphism groups of these three families of graphs, and Feng [9] generalized this result by proving that for any minimal generating set S of transpositions of Sn , the Cayley graph Cay(Sn , S ) is normal, that is, Aut(Sn , S ) = R(Sn )oAut(Sn , S ). For more results regarding automorphism groups of Cayley graphs, we refer the reader to [10–13,1]. We also refer the reader to [14–16] for the applications of Cayley graphs to interconnection networks. In this article, we completely determine the full automorphism group of alternating group graphs. The following is the main result. Theorem. Let AGn be the alternating group graph defined in (1). Then Aut(AGn ) = R(An ) o Aut(An , S ). Furthermore, Aut(An , S ) = ⟨σ ((1 2))⟩ × ⟨σ ((3 4)), σ ((3 5)), . . . , σ ((3 n))⟩ ∼ = S2 × Sn−2 , where σ (g ) is the automorphism of An induced by conjugacy action of g for g ∈ An . Remark. 1. In [17], Huan et al. also considered the automorphism group of AGn . They proved that Aut(AGn ) ∼ = R(An ) o Sn−1 . However, with the help of computer software package MAGMA [18], it can be obtained that Aut(AG4 ) ∼ = R(A4 ) o (Z2 × Z2 ). 2. As a consequence of the theorem, we have that AGn is a normal Cayley graph which is arc-transitive but not 2-arctransitive. 3. Let H = ⟨σ ((1 2))⟩ × ⟨σ ((3 4 . . . n))⟩. It is easily seen that H ≤ Aut(An , S ) acts regularly on S. From [16] we know that for the alternating group graph there exists a gossiping protocol which exhibits very attractive features essential for efficient communication, and an algorithm for constructing such a protocol is also given. See [16, Theorem 4.1, Algorithm 4.3 and Corollary 4.4] for details. 4. Let G be a connected graph which is neither a star nor a triangle. The restricted edge-connectivity λ′ (G) of G is defined to be the minimum cardinality over all restricted edge-cuts of G. If λ′ (G) equals the minimum edge-degree of G, then G is said to be optimal. An optimal graph G is said to be super restricted edge-connected, for short, super-λ′ , if every minimum restricted edge-cut of G isolates an edge. In [19] it is shown that a connected edge-transitive graph G of valency k is not super-λ′ if and only if either G is isomorphic to the three-dimensional hypercube Q3 , or |G| ≥ 6 and (k, g ) = (4, 3), where g is the girth of G. Note that |AGn | ≥ 12 for each n ≥ 4, and that AGn is edge-transitive because it is arc-transitive by the theorem. This implies that the alternating group graph AGn is super-λ′ for each n ≥ 4. 2. Proof of the theorem Let A = Aut(AGn ) and let Ae be the stabilizer of the identity e of An in A. If n = 3, then AG3 = Cay(A3 , {(1 2 3), (1 3 2)}) is a triangle and A = R(A3 ) o ⟨σ ((1 2))⟩, as required. In what follows we assume that n ≥ 4. We first prove a claim. 1 Claim. Let s1 = (1 2 i) and s2 = (1 2 j) with i ̸= j. Then C1 = (e, s1 , s2 s1 , s− 2 ) is the unique 4-cycle in AGn passing through −1 −1 e, s1 and s2 , and C2 = (s1 , s2 s1 , s2 s1 ) is the unique triangle in AGn passing through s1 and s2 s1 . 1 −1 ′ It is easy to check that C1 = (e, s1 , s2 s1 , s− 2 ) is a 4-cycle. Assume that C1 is another 4-cycle passing through e, s1 and s2 . ′ −1 Then C1 = (e, (1 2 i), x(1 2 i) = y(1 j 2), (1 j 2)) for some x, y ∈ S. It follows that x y = (1 2 i) (1 2 j) = (1 j) (2 i). As x, y ∈ S, one has x−1 = (1 2 a) or (1 a 2) and y = (1 2 b) or (1 b 2) for some integers 3 ≤ a, b ≤ n. Since x−1 y is an involution, a ̸= b and either x−1 = (1 2 a) and y = (1 2 b) or x−1 = (1 a 2) and y = (1 b 2). For the former, (1 j) (2 i) = x−1 y = (1 b) (2 a), implying that a = i and b = j. It follows that x = (1 i 2) and hence x(1 2 i) = e, a contradiction. Thus, x−1 = (1 a 2) and y = (1 b 2), and hence (1 j) (2 i) = x−1 y = (1 a) (2 b). It follows that a = j and b = i, and hence x = (1 2 j) and x(1 2 i) = (1 i) (2 j), implying that C1′ = (e, (1 2 i), (1 i) (2 j), (1 j 2)) = C1 . 1 −1 ′ Since s− 2 is of order 3, C2 = (s1 , s2 s1 , s2 s1 ) is a triangle. Let C2 be an arbitrary triangle passing through s1 and s2 s1 . Then ′ C2 = ((1 2 i), u(1 2 i) = v(1 i) (2 j), (1 i) (2 j)) for some u, v ∈ S. It follows that u−1 v = (1 2 i)[(1 i) (2 j)] = (1 j 2). Since u−1 , v ∈ S, one has u−1 = v = (1 2 j), implying C2′ = ((1 2 i), (1 j i), (1 i) (2 j)) = C2 . Now we know that the claim is true. Let A∗e be the kernel of Ae acting on S. Take s ∈ S. Then s = (1 2 i) or (1 i 2) for some 3 ≤ i ≤ n. We shall show that A∗e fixes every neighbor of s.
J.-X. Zhou / Applied Mathematics Letters 24 (2011) 229–231
231
Let s = (1 2 i). The neighborhood of s is {e, (1 i 2), (1 i) (2 j), (1 j i) | 3 ≤ j ≤ n and j ̸= i}. Clearly, A∗e fixes e. Also, A∗e fixes (1 i 2) because (1 i 2) ∈ S. Let 3 ≤ j ≤ n and j ̸= i. By the claim, C1 = (e, (1 2 i), (1 i) (2 j), (1 j 2)) is the unique 4-cycle passing through e, (1 2 i) and (1 j 2). Then A∗e fixes (1 i) (2 j) because (1 2 i), (1 j 2) ∈ S. This implies that A∗e also fixes (1 j i) because C2 = ((1 2 i), (1 j i), (1 i) (2 j)) is the unique triangle passing through (1 2 i) and (1 i) (2 j) by the claim. Let s = (1 i 2). The neighborhood of s is {e, (1 2 i), (1 j) (2 i), (2 j i) | 3 ≤ j ≤ n and j ̸= i}. Clearly, A∗e fixes e and (1 i 2). It is easy to see that σ ((1 2)), the automorphism of An induced by the conjugacy action of (1 2), is an automorphism of AGn . Let 3 ≤ j ≤ n and j ̸= i. Since (1 i 2)σ (1 2) = (1 2 i) and (1 j 2)σ (1 2) = (1 2 j), by the claim, C3 = (e, (1 i 2), (1 j) (2 i), (1 2 j)) is the unique 4-cycle in AGn passing through (1 i 2) and (1 2 j), and C4 = ((1 i 2), (2 j i), (1 j) (2 i)) is the unique triangle in AGn passing through (1 i 2) and (1 j) (2 i). As (1 i 2), (1 2 j) ∈ S, A∗e fixes (1 j) (2 i) and (2 j i). Now we can conclude that A∗e fixes all neighbors of s for any s ∈ S. The connectivity of AGn implies that A∗e fixes all vertices of AGn , and hence A∗e = 1. Then Ae = Ae /A∗e can be viewed as a permutation group on S. For any 3 ≤ i ≤ n, set ∆i = {(1 2 i), (1 i 2)}. It is easy to see that there are exactly n−2 triangles in AGn passing through e which are (e, (1 2 i), (1 i 2)) with 3 ≤ i ≤ n. This implies that ∆i is an imprimitive block of Ae on S. Set ∆ = {∆i | 3 ≤ i ≤ n}. Then, |∆| = n − 2, and Ae acts on ∆. Let K be the kernel of this action. Then Ae /K ≤ Sn−2 . It is clear that K acts on ∆3 . Let K ∗ be the kernel of this action. For any 3 < i ≤ n, by the claim, (e, (1 2 3), (1 3) (2 i), (1 i 2)) is a 4-cycle in AGn passing through (1 2 3) and (1 i 2). It is easy to see that except for e, (1 2 3) and (1 2 i) have no other common neighbors. Hence there is no 4-cycle in AGn passing through (1 2 3) and (1 2 i). Since K ∗ fixes ∆i = {(1 2 i), (1 i 2)} setwise, K ∗ must fix (1 2 i) and (1 i 2). By the arbitrariness of i, K ∗ fixes every element in S, implying that K ∗ ≤ A∗e = 1. It follows that K ≤ Z2 because |∆3 | = 2. Thus, |Ae | = |K | |Ae /K | ≤ 2|Sn−2 |. For any g ∈ Sn , let σ (g ) denote the automorphism of An induced by the conjugacy action of g. It is easy to check that σ ((1 2)) and σ ((3 i)) (4 ≤ i ≤ n) are in Aut(An , S ). Furthermore, by the elementary group theory,
⟨σ ((1 2))⟩ × ⟨σ ((3 4)), σ ((3 5)), . . . , σ ((3 n))⟩ ∼ = S2 × Sn−2 . Since |Aut(An , S )| ≤ |Ae | ≤ 2|Sn−2 |, it follows that Aut(An , S ) = ⟨σ ((1 2))⟩ × ⟨σ ((3 4)), σ ((3 5)), . . . , σ ((3 n))⟩, and hence A = R(An ) o Aut(An , S ).
Acknowledgements This work was supported by the National Natural Science Foundation of China (10901015, 10871021) and the Science and Technology Foundation of Beijing Jiaotong University (2008RC037). The author is indebted to the anonymous referees for many valuable comments and constructive suggestions. References [1] M.-Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998) 309–319. [2] J.-S. Jwo, S. Lakshmivarahan, S.K. Dhall, A new class of interconnection networks based on the alternating group, Networks 23 (1993) 315–326. [3] J.-M. Chang, J.-S. Yang, Panconnectivity, fault tolerant Hamiltonicity and Hamiltonian connectivity in alternating group graphs, Networks 44 (2004) 302–310. [4] J.-M. Chang, J.-S. Yang, Fault-tolerant cycle-embedding in alternating group graphs, Appl. Math. Comput. 197 (2008) 760–767. [5] Y.-H. Teng, Jimmy J.-M. Tan, L.-H. Hsu, Panpositionable Hamiltonicity of the alternating group graphs, Networks 50 (2007) 146–156. [6] S. Lakshmivarahan, J.-S. Jwo, S.K. Dhall, Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey, Parallel Comput. 19 (1993) 361–407. [7] Q.-X. Huang, Z. Zhang, On the Cayley graphs of Sym(n) with respect to transposition connection (submitted for publication). [8] Z. Zhang, Q.-X. Huang, Automorphism group of bubble-sort graphs and modified bubble-sort graphs, Adv. Math. 34 (2005) 441–447. [9] Y.-Q. Feng, Automorphism groups of Cayley graphs on symmetric groups with generating transposition sets, J. Combin. Theory B 96 (2006) 67–72. [10] Y.-Q. Feng, Z.-P. Lu, M.-Y. Xu, Automorphism groups of Cayley digraphs, in: J. Koolen, J.H. Kwak, M.Y. Xu (Eds.), Application of Group Theory to Combinatorics, Taylor & Francis Group, London, 2008, pp. 13–25. [11] C.D. Godsil, The automorphism groups of some cubic Cayley graphs, European J. Combin. 4 (1983) 25–32. [12] C.H. Li, On isomorphisms of connected Cayley graphs III, Bull. Austral. Math. Soc. 58 (1998) 137–145. [13] C.H. Li, The solution of a problem of Godsil on cubic Cayley graphs, J. Combin. Theory B 72 (1998) 140–142. [14] M.-C. Heydemann, Cayley graphs and interconnection networks, in: G. Hahn, G. Sabidussi (Eds.), Graph Symmetry, Kluwer Academic Publishing, Dordrecht, 1997, pp. 167–224. [15] A. Thomson, S. Zhou, Frobenius circulant graphs of valency four, J. Aust. Math. Soc. 85 (2008) 269–282. [16] S. Zhou, A class of arc-transitive Cayley graphs as models for interconnection networks, SIAM J. Discrete Math. 23 (2009) 694–714. [17] H.-L. Huan, H.-M. Lei, W. Xie, Automorphism groups of a family of Cayley graphs on alternating groups, J. Syst. Sci. Inform. 5 (2007) 37–42. [18] W. Bosma, C. Cannon, C. Playoust, The MAGMA algebra system I: the user language, J. Symb. Comput. 24 (1997) 235–265. [19] J.-X. Zhou, Super restricted edge-connectivity of regular edge-transitive graphs (submitted for publication).
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
The automorphism group of the alternating group graph Jin-Xin Zhou Department of Mathematics, Beijing Jiaotong University, Beijing 100044, PR China
article
info
Article history: Received 2 May 2009 Received in revised form 14 September 2010 Accepted 15 September 2010 Keywords: Alternating group graph Automorphism group Cayley graph
abstract Let An be the alternating group of degree n with n ≥ 3. Set S = {(1 2 i), (1 i 2) | 3 ≤ i ≤ n}. The alternating group graph, denoted by AGn , is defined as the Cayley graph on An with respect to S. Jwo et al. (1993) [J.-S. Jwo, S. Lakshmivarahan, S.K. Dhall, A new class of interconnection networks based on the alternating group, Networks 23 (1993) 315–326] introduced the alternating group graph AGn as an interconnection network topology for computing systems, and they proved that AGn is arc-transitive. In this work, it is shown that the full automorphism group of AGn is the semi-direct product R(An ) o Aut(An , S ), where R(An ) is the right regular representation of An and Aut(An , S ) = {α ∈ Aut(An ) | S α = S } ∼ = Sn−2 × S2 . It follows from this result that AGn is arc-transitive but not 2-arc-transitive. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Graphs considered in the work are finite, connected, simple and undirected. For a graph X , denote by Aut(X ) the full automorphism group of X . An s-arc in a graph X is an ordered (s + 1)-tuple (v0 , v1 , . . . , vs−1 , vs ) of vertices of X such that vi−1 is adjacent to vi for 1 ≤ i ≤ s and vi−1 ̸= vi+1 for 1 ≤ i ≤ s − 1. A graph X is said to be s-arc-transitive if Aut(X ) is transitive on the set of s-arcs in X . In particular, 0-arc-transitivity means vertex-transitivity, and 1-arc-transitivity means arc-transitivity. For a set V and a group G with identity element 1, an action of G on V is a mapping V × G → V , (v, g ) → v g , such that v 1 = v and (v g )h = v gh for v ∈ V and g , h ∈ G. The subgroup K = {g ∈ G | v g = v, ∀v ∈ V } of G is called the kernel of G acting on V . For two groups K , H, if H acts on K (as a set) such that (xy)h = xh yh for any x, y ∈ K and h ∈ H, then H is said to act on K as a group. In this case, we use K o H to denote the semi-direct product of K with H with respect to the action. For a finite group G and a subset S of G such that 1 ̸∈ S and S = S −1 (where 1 is the identity element of G), the Cayley graph Cay(G, S ) on G with respect to S is defined to have vertex set G and edge set {{g , sg } | g ∈ G, s ∈ S }. The automorphism group Aut(Cay(G, S )) of Cay(G, S ) contains the right regular representation R(G) of G, the action of G on itself by right multiplication, as a subgroup. Thus Cay(G, S ) is vertex-transitive. Furthermore, Aut(G, S ) = {α ∈ Aut(G) | S α = S } is a subgroup of Aut(Cay(G, S ))1 , the stabilizer of the vertex 1 in Aut(Cay(G, S )). A Cayley graph Cay(G, S ) is said to be normal if R(G) is normal in Aut(Cay(G, S )). Xu [1, Proposition 1.5] proved that Cay(G, S ) is normal if and only if Aut(Cay(G, S ))1 = Aut(G, S ). Let An be the alternating group of degree n with n ≥ 3. Set S = {(1 2 i), (1 i 2) | 3 ≤ i ≤ n}. The alternating group graph, denoted by AGn , is defined as the Cayley graph AGn = Cay(An , S ).
(1)
Jwo et al. [2] introduced the alternating group graph as an interconnection network topology for computing systems. Following this pioneering article, the alternating group graph has been extensively studied over decades by many authors. For example, Chang and Yang [3] investigated the panconnectivity, and Hamiltonian-connectivity in alternating group
E-mail address: [email protected]. 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.09.009
230
J.-X. Zhou / Applied Mathematics Letters 24 (2011) 229–231
graphs, and they [4] also studied the fault-tolerant Hamiltonicity of alternating group graphs. Teng et al. [5] proved that if n ≥ 3 then for any two different vertices x and y of AGn and for any integer ℓ satisfying d(x, y) ≤ ℓ ≤ |AGn | − d(x, y), there exists a Hamiltonian cycle C of AGn such that the relative distance between x, y on C is ℓ, where d(x, y) is the distance between x and y in AGn . It is known from [2] that AGn is arc-transitive. In many situations, it is highly advantageous to use interconnection networks which are highly symmetric. This often simplifies the computational and routing algorithms. When dealing with the symmetry of graphs, the goal is to gain as much information as possible about structure properties of full automorphism groups of graphs. Recently, several publications have been focused on investigation of automorphism groups of Cayley graphs having connection with interconnection networks. For example, let Sn be the symmetric group of degree n, and set T1 = {(i i + 1)|1 ≤ i ≤ n − 1}, T2 = T1 ∪ {(1 n)} and T3 = {(1 i)|2 ≤ i ≤ n}. The Cayley graphs Cay(Sn , Tj )(1 ≤ j ≤ 3) are called the bubble-sort graph, modified bubble-sort graph and star graph, respectively, in [6]. Huang and Zhang [7,8] determined the full automorphism groups of these three families of graphs, and Feng [9] generalized this result by proving that for any minimal generating set S of transpositions of Sn , the Cayley graph Cay(Sn , S ) is normal, that is, Aut(Sn , S ) = R(Sn )oAut(Sn , S ). For more results regarding automorphism groups of Cayley graphs, we refer the reader to [10–13,1]. We also refer the reader to [14–16] for the applications of Cayley graphs to interconnection networks. In this article, we completely determine the full automorphism group of alternating group graphs. The following is the main result. Theorem. Let AGn be the alternating group graph defined in (1). Then Aut(AGn ) = R(An ) o Aut(An , S ). Furthermore, Aut(An , S ) = ⟨σ ((1 2))⟩ × ⟨σ ((3 4)), σ ((3 5)), . . . , σ ((3 n))⟩ ∼ = S2 × Sn−2 , where σ (g ) is the automorphism of An induced by conjugacy action of g for g ∈ An . Remark. 1. In [17], Huan et al. also considered the automorphism group of AGn . They proved that Aut(AGn ) ∼ = R(An ) o Sn−1 . However, with the help of computer software package MAGMA [18], it can be obtained that Aut(AG4 ) ∼ = R(A4 ) o (Z2 × Z2 ). 2. As a consequence of the theorem, we have that AGn is a normal Cayley graph which is arc-transitive but not 2-arctransitive. 3. Let H = ⟨σ ((1 2))⟩ × ⟨σ ((3 4 . . . n))⟩. It is easily seen that H ≤ Aut(An , S ) acts regularly on S. From [16] we know that for the alternating group graph there exists a gossiping protocol which exhibits very attractive features essential for efficient communication, and an algorithm for constructing such a protocol is also given. See [16, Theorem 4.1, Algorithm 4.3 and Corollary 4.4] for details. 4. Let G be a connected graph which is neither a star nor a triangle. The restricted edge-connectivity λ′ (G) of G is defined to be the minimum cardinality over all restricted edge-cuts of G. If λ′ (G) equals the minimum edge-degree of G, then G is said to be optimal. An optimal graph G is said to be super restricted edge-connected, for short, super-λ′ , if every minimum restricted edge-cut of G isolates an edge. In [19] it is shown that a connected edge-transitive graph G of valency k is not super-λ′ if and only if either G is isomorphic to the three-dimensional hypercube Q3 , or |G| ≥ 6 and (k, g ) = (4, 3), where g is the girth of G. Note that |AGn | ≥ 12 for each n ≥ 4, and that AGn is edge-transitive because it is arc-transitive by the theorem. This implies that the alternating group graph AGn is super-λ′ for each n ≥ 4. 2. Proof of the theorem Let A = Aut(AGn ) and let Ae be the stabilizer of the identity e of An in A. If n = 3, then AG3 = Cay(A3 , {(1 2 3), (1 3 2)}) is a triangle and A = R(A3 ) o ⟨σ ((1 2))⟩, as required. In what follows we assume that n ≥ 4. We first prove a claim. 1 Claim. Let s1 = (1 2 i) and s2 = (1 2 j) with i ̸= j. Then C1 = (e, s1 , s2 s1 , s− 2 ) is the unique 4-cycle in AGn passing through −1 −1 e, s1 and s2 , and C2 = (s1 , s2 s1 , s2 s1 ) is the unique triangle in AGn passing through s1 and s2 s1 . 1 −1 ′ It is easy to check that C1 = (e, s1 , s2 s1 , s− 2 ) is a 4-cycle. Assume that C1 is another 4-cycle passing through e, s1 and s2 . ′ −1 Then C1 = (e, (1 2 i), x(1 2 i) = y(1 j 2), (1 j 2)) for some x, y ∈ S. It follows that x y = (1 2 i) (1 2 j) = (1 j) (2 i). As x, y ∈ S, one has x−1 = (1 2 a) or (1 a 2) and y = (1 2 b) or (1 b 2) for some integers 3 ≤ a, b ≤ n. Since x−1 y is an involution, a ̸= b and either x−1 = (1 2 a) and y = (1 2 b) or x−1 = (1 a 2) and y = (1 b 2). For the former, (1 j) (2 i) = x−1 y = (1 b) (2 a), implying that a = i and b = j. It follows that x = (1 i 2) and hence x(1 2 i) = e, a contradiction. Thus, x−1 = (1 a 2) and y = (1 b 2), and hence (1 j) (2 i) = x−1 y = (1 a) (2 b). It follows that a = j and b = i, and hence x = (1 2 j) and x(1 2 i) = (1 i) (2 j), implying that C1′ = (e, (1 2 i), (1 i) (2 j), (1 j 2)) = C1 . 1 −1 ′ Since s− 2 is of order 3, C2 = (s1 , s2 s1 , s2 s1 ) is a triangle. Let C2 be an arbitrary triangle passing through s1 and s2 s1 . Then ′ C2 = ((1 2 i), u(1 2 i) = v(1 i) (2 j), (1 i) (2 j)) for some u, v ∈ S. It follows that u−1 v = (1 2 i)[(1 i) (2 j)] = (1 j 2). Since u−1 , v ∈ S, one has u−1 = v = (1 2 j), implying C2′ = ((1 2 i), (1 j i), (1 i) (2 j)) = C2 . Now we know that the claim is true. Let A∗e be the kernel of Ae acting on S. Take s ∈ S. Then s = (1 2 i) or (1 i 2) for some 3 ≤ i ≤ n. We shall show that A∗e fixes every neighbor of s.
J.-X. Zhou / Applied Mathematics Letters 24 (2011) 229–231
231
Let s = (1 2 i). The neighborhood of s is {e, (1 i 2), (1 i) (2 j), (1 j i) | 3 ≤ j ≤ n and j ̸= i}. Clearly, A∗e fixes e. Also, A∗e fixes (1 i 2) because (1 i 2) ∈ S. Let 3 ≤ j ≤ n and j ̸= i. By the claim, C1 = (e, (1 2 i), (1 i) (2 j), (1 j 2)) is the unique 4-cycle passing through e, (1 2 i) and (1 j 2). Then A∗e fixes (1 i) (2 j) because (1 2 i), (1 j 2) ∈ S. This implies that A∗e also fixes (1 j i) because C2 = ((1 2 i), (1 j i), (1 i) (2 j)) is the unique triangle passing through (1 2 i) and (1 i) (2 j) by the claim. Let s = (1 i 2). The neighborhood of s is {e, (1 2 i), (1 j) (2 i), (2 j i) | 3 ≤ j ≤ n and j ̸= i}. Clearly, A∗e fixes e and (1 i 2). It is easy to see that σ ((1 2)), the automorphism of An induced by the conjugacy action of (1 2), is an automorphism of AGn . Let 3 ≤ j ≤ n and j ̸= i. Since (1 i 2)σ (1 2) = (1 2 i) and (1 j 2)σ (1 2) = (1 2 j), by the claim, C3 = (e, (1 i 2), (1 j) (2 i), (1 2 j)) is the unique 4-cycle in AGn passing through (1 i 2) and (1 2 j), and C4 = ((1 i 2), (2 j i), (1 j) (2 i)) is the unique triangle in AGn passing through (1 i 2) and (1 j) (2 i). As (1 i 2), (1 2 j) ∈ S, A∗e fixes (1 j) (2 i) and (2 j i). Now we can conclude that A∗e fixes all neighbors of s for any s ∈ S. The connectivity of AGn implies that A∗e fixes all vertices of AGn , and hence A∗e = 1. Then Ae = Ae /A∗e can be viewed as a permutation group on S. For any 3 ≤ i ≤ n, set ∆i = {(1 2 i), (1 i 2)}. It is easy to see that there are exactly n−2 triangles in AGn passing through e which are (e, (1 2 i), (1 i 2)) with 3 ≤ i ≤ n. This implies that ∆i is an imprimitive block of Ae on S. Set ∆ = {∆i | 3 ≤ i ≤ n}. Then, |∆| = n − 2, and Ae acts on ∆. Let K be the kernel of this action. Then Ae /K ≤ Sn−2 . It is clear that K acts on ∆3 . Let K ∗ be the kernel of this action. For any 3 < i ≤ n, by the claim, (e, (1 2 3), (1 3) (2 i), (1 i 2)) is a 4-cycle in AGn passing through (1 2 3) and (1 i 2). It is easy to see that except for e, (1 2 3) and (1 2 i) have no other common neighbors. Hence there is no 4-cycle in AGn passing through (1 2 3) and (1 2 i). Since K ∗ fixes ∆i = {(1 2 i), (1 i 2)} setwise, K ∗ must fix (1 2 i) and (1 i 2). By the arbitrariness of i, K ∗ fixes every element in S, implying that K ∗ ≤ A∗e = 1. It follows that K ≤ Z2 because |∆3 | = 2. Thus, |Ae | = |K | |Ae /K | ≤ 2|Sn−2 |. For any g ∈ Sn , let σ (g ) denote the automorphism of An induced by the conjugacy action of g. It is easy to check that σ ((1 2)) and σ ((3 i)) (4 ≤ i ≤ n) are in Aut(An , S ). Furthermore, by the elementary group theory,
⟨σ ((1 2))⟩ × ⟨σ ((3 4)), σ ((3 5)), . . . , σ ((3 n))⟩ ∼ = S2 × Sn−2 . Since |Aut(An , S )| ≤ |Ae | ≤ 2|Sn−2 |, it follows that Aut(An , S ) = ⟨σ ((1 2))⟩ × ⟨σ ((3 4)), σ ((3 5)), . . . , σ ((3 n))⟩, and hence A = R(An ) o Aut(An , S ).
Acknowledgements This work was supported by the National Natural Science Foundation of China (10901015, 10871021) and the Science and Technology Foundation of Beijing Jiaotong University (2008RC037). The author is indebted to the anonymous referees for many valuable comments and constructive suggestions. References [1] M.-Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998) 309–319. [2] J.-S. Jwo, S. Lakshmivarahan, S.K. Dhall, A new class of interconnection networks based on the alternating group, Networks 23 (1993) 315–326. [3] J.-M. Chang, J.-S. Yang, Panconnectivity, fault tolerant Hamiltonicity and Hamiltonian connectivity in alternating group graphs, Networks 44 (2004) 302–310. [4] J.-M. Chang, J.-S. Yang, Fault-tolerant cycle-embedding in alternating group graphs, Appl. Math. Comput. 197 (2008) 760–767. [5] Y.-H. Teng, Jimmy J.-M. Tan, L.-H. Hsu, Panpositionable Hamiltonicity of the alternating group graphs, Networks 50 (2007) 146–156. [6] S. Lakshmivarahan, J.-S. Jwo, S.K. Dhall, Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey, Parallel Comput. 19 (1993) 361–407. [7] Q.-X. Huang, Z. Zhang, On the Cayley graphs of Sym(n) with respect to transposition connection (submitted for publication). [8] Z. Zhang, Q.-X. Huang, Automorphism group of bubble-sort graphs and modified bubble-sort graphs, Adv. Math. 34 (2005) 441–447. [9] Y.-Q. Feng, Automorphism groups of Cayley graphs on symmetric groups with generating transposition sets, J. Combin. Theory B 96 (2006) 67–72. [10] Y.-Q. Feng, Z.-P. Lu, M.-Y. Xu, Automorphism groups of Cayley digraphs, in: J. Koolen, J.H. Kwak, M.Y. Xu (Eds.), Application of Group Theory to Combinatorics, Taylor & Francis Group, London, 2008, pp. 13–25. [11] C.D. Godsil, The automorphism groups of some cubic Cayley graphs, European J. Combin. 4 (1983) 25–32. [12] C.H. Li, On isomorphisms of connected Cayley graphs III, Bull. Austral. Math. Soc. 58 (1998) 137–145. [13] C.H. Li, The solution of a problem of Godsil on cubic Cayley graphs, J. Combin. Theory B 72 (1998) 140–142. [14] M.-C. Heydemann, Cayley graphs and interconnection networks, in: G. Hahn, G. Sabidussi (Eds.), Graph Symmetry, Kluwer Academic Publishing, Dordrecht, 1997, pp. 167–224. [15] A. Thomson, S. Zhou, Frobenius circulant graphs of valency four, J. Aust. Math. Soc. 85 (2008) 269–282. [16] S. Zhou, A class of arc-transitive Cayley graphs as models for interconnection networks, SIAM J. Discrete Math. 23 (2009) 694–714. [17] H.-L. Huan, H.-M. Lei, W. Xie, Automorphism groups of a family of Cayley graphs on alternating groups, J. Syst. Sci. Inform. 5 (2007) 37–42. [18] W. Bosma, C. Cannon, C. Playoust, The MAGMA algebra system I: the user language, J. Symb. Comput. 24 (1997) 235–265. [19] J.-X. Zhou, Super restricted edge-connectivity of regular edge-transitive graphs (submitted for publication).