J Supercomput (2012) 59:246–248DOI 10.1007/s11227-010-0434-y

A note on the alternating group network

Eddie Cheng · Ke Qiu · Zhizhang Shen

Published online: 16 April 2010© Springer Science+Business Media, LLC 2010

Abstract The class of alternating group networks was introduced in the late 1990’sas an alternative to the alternating group graphs as interconnection networks. Re-cently, additional properties for the alternating group networks have been published.In particular, Zhou et al., J. Supercomput (2009), doi:10.1007/s11227-009-0304-7,was published very recently in this journal. We show that this so-called new inter-connection topology is in fact isomorphic to the (n,n − 2)-star, a member of thewell-known (n, k)-stars, 1 ≤ k ≤ n − 1, a class of popular networks proposed earlierfor which a large amount of work have already been done. Specifically, the problem inZhou et al., J. Supercomput (2009), doi:10.1007/s11227-009-0304-7, was addressedin Lin and Duh, Inf. Sci. 178(3), 788–801, 2008, when k = n − 2.

Keywords Alternating group network · (n, k)-star graph · Isomorphism

1 ANn is isomorphic to (n,n − 2)-star

The alternating group network [6] was introduced as an alternative to the alternat-ing group graphs [7]. Supposing n ≥ 3, the alternating group network ANn has the

E. ChengDept. of Mathematics and Statistics, Oakland University, Rochester, MI 48309-4401, USAe-mail: echeng@oakland.edu

K. Qiu (�)Department of Computer Science, Brock University, St. Catharines, Ontario L2S 3A1, Canadae-mail: kqiu@brocku.ca

Z. ShenDept. of Computer Science and Technology, Plymouth State University, Plymouth, NH 03264-1595,USAe-mail: zshen@plymouth.edu

A note on the alternating group network 247

vertex set of even permutations from {1,2, . . . , n}, two vertices [a1, a2, a3, . . . , an]and [b1, b2, b3, . . . , bn] are adjacent if one of the following three conditions is sat-isfied. The first is that there exists an i ∈ {4,5, . . . , n} such that a1 = b2, a2 = b1,a3 = bi , ai = b3 and aj = bj for j ∈ {4,5, . . . , n} \ {i}. The second is a1 = b2,a2 = b3, a3 = b1 and aj = bj for 4 ≤ j ≤ n. The third is a1 = b3, a2 = b1, a3 = b2

and aj = bj for 4 ≤ j ≤ n.Since its proposal, the alternating group network has attracted some attention and

research has been done in [2, 3]. In [2], the network is studied in terms of node-to-node distance and optimal routing, while the Whitney numbers of the second kind,also known as the surface areas, are studied in [3]. Most recently, vertex-disjoint pathsare constructed for this network in this very journal [9].

In this short note, we would like to point out that this so-called new interconnectiontopology is in fact isomorphic to the well-known (n, k)-star, proposed earlier [4, 5]to address deficiency (most notably the scalability) in the star network [1]. Becausethe alternating group network is a special case of the (n, k)-star graph, many resultsare already known and they need not be studied separately. For example, the sameproblem addressed in [9] was studied in [8] when k = n − 2.

The (n, k)-star has been studied extensively in areas such as broadcasting andsorting, distance formula and shortest distance routing, fault tolerant routings, super-connectedness, vulnerability, Hamiltonianicity and various generalizations.

Suppose n ≥ 3 and 2 ≤ k ≤ n − 1. The vertex set of an (n, k)-star graphis the k-permutations from {1,2, . . . , n}, and two vertices [a1, a2, a3, . . . , ak] and[b1, b2, b3, . . . , bk] are adjacent if one of the following two conditions is satisfied.The first is there exists an i ∈ {2,3, . . . , k} such that a1 = bi , ai = b1 and aj = bj forj ∈ {1,2, . . . , k} \ {1, i} (that is, one is obtained from the other by switching the sym-bols in the first and the ith position). The second is a1 �= b1 and aj = bj for 2 ≤ j ≤ k

(that is, one is obtained from the other by replacing the symbol in the first positionby a symbol not in the permutation).

Theorem 1 Let n ≥ 3. Then ANn is isomorphic to (n,n − 2)-star.

Proof Consider a vertex π = [a1, a2, . . . , an−2] in (n,n − 2)-star. We will extendthis to a full even permutation πE on {1,2, . . . , n}. Let {y, z} = {1,2, . . . , n} \{a1, a2, . . . , an−2}. Then exactly one of [y, z, a1, a2, . . . ,

an−2] and [z, y, a1, a2, . . . , an−2] is even. Let πE be the even permutation. Weclaim that φ : V (Sn,n−2) −→ V (ANn), defined by φ(π) = πE , is an isomor-phism. Consider π = [a1, a2, . . . , an−2] and πE = [y, z, a1, a2, . . . , an−2]. Let α =[b1, b2, . . . , bn] be obtained from π by switching the symbols in the first and theith position. Then αE = [z, y, b1, b2, . . . , bn] as we want αE to be even. This isprecisely Condition 1 for ANn. Now suppose we replace a1 by y in π to obtainα = [y, a2, . . . , an]. Then αE is either [z, a1, y, a2, . . . , an] or [a1, z, y, a2, . . . , an].Since we need to choose the even one, αE = [z, a1, y, a2, . . . , an]. But this is Condi-tion 2 for ANn. Similarly, if we replace a1 by z in π to obtain α, then πE and αE arerelated as in Condition 3 for ANn. �

248 E. Cheng et al.

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