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Cluster Comput (2011) 14:483–490DOI 10.1007/s10586-011-0189-0
A new family of Cayley graph interconnection networks basedon wreath product and its topological properties
Zhen Zhang · Wenjun Xiao
Received: 14 May 2010 / Accepted: 10 October 2011 / Published online: 4 November 2011© Springer Science+Business Media, LLC 2011
Abstract This paper introduces a new type of Cayleygraphs for building large-scale interconnection networks,namely WG2m
n , whose vertex degree is m + 3 when n ≥ 3and is m + 2 when n = 2. A routing algorithm for the pro-posed graph is also presented, and the upper bound of thediameter is deduced as �5n/2�. Moreover, the embeddingproperties and maximal fault tolerance are also analyzed.Finally, we compare the proposed networks with some othersimilar network topologies. It is found that WG2m
n is supe-rior to other interconnection networks because it helps toconstruct large-scale networks with lower cost.
Keywords Interconnection networks · Cayley graph ·Routing algorithm · Diameter · Embedding · Fault tolerance
1 Introduction
Desirable topological properties of an interconnection net-work include low degree, low diameter, symmetry, high con-nectivity and high fault tolerance. For the past several years,there has been active research on a class of graphs calledCayley graphs because these graphs possess many of theabove properties.
Let G be a group and S a subset of G. The subset S is saidto be the generator set for G, and the elements of S are called
Z. Zhang (�)Department of Computer Science, Jinan University, Guangzhou,Chinae-mail: [email protected]
W. XiaoSchool of Computer Science and Engineering, South ChinaUniversity of Technology, Guangzhou, Chinae-mail: [email protected]
generators of G, if every element of G can be expressed asa finite product of their powers. The Cayley graph of thegroup G and generator S, denoted by Cay(G,S), has ver-tices that are elements of G and arcs that are ordered pairs(g,gs) for g ∈ S, s ∈ S. When identify element e /∈ S andS = S−1, the graph Cay(G,S) is a simple graph. Every Cay-ley graph is regular and vertex transitive [1, 3, 5]. That is tosay, for every pair of vertices u and v, there exists an auto-morphism of the graph that maps u in v. An attractive fea-ture of vertex-transitive graphs is that routing between twoarbitrary vertices reduces to routing from an arbitrary vertexto the identity vertex.
A class of Cayley graphs based on permutation groupshas been proven to be suitable for designing interconnectionnetworks, including Star graph [1, 2, 14], Hypercubes [4],Pancake graphs [2, 14]. However the vertex degree in-creases with the number vertices so that using these graphs isprohibitive in networks with large number of vertices. Thereare many applications where the computing vertices in aninterconnection network only have a fixed number of I/Oports. Moreover, fixed degree networks are important fromthe viewpoint of VLSI implementation [15].
Vadapalli and Srimani proposed the trivalent Cayleygraph with a fixed degree of three [17]. This class of graphsis known to have logarithmic diameter and maximal faulttolerance [13, 16–19]. Then, they proposed a family of Cay-ley graph of constant degree four, which is isomorphic to thewrapped around butterfly graph [18, 20]. Fu and Chau pro-posed cylic-cubes, which are Cayley graphs with fixed de-grees being any even number greater than or equal four [10].These graphs have optimal fault tolerance and logarithmicdiameters. The shortest path routing and embedding of aHamiltonian cycle, meshes, and hypercubes are also dis-cussed in [10]. New classes of interconnection networkswith odd fix degrees are also developed in [7, 22]. Hsieh
484 Cluster Comput (2011) 14:483–490
and Hsiao develop a new family Cayley graph Gk,n withk-degree, which possess a useful property in that the degreeof each vertex is fixed by a general positive integer k withoutregard to the number of vertices [12].
The scale of some interconnection networks has beenincreasing dramatically in recent years, such as supercom-puter, VLSI and UVSI. Modern supercomputers fulfill theirpromises of higher performance by fully exploiting the ca-pabilities of tens of thousands of processors interconnectedwithin an interconnection network. The fastest supercom-puter is K Computer in Japan [23]. The system is still un-der construction and will enter service in 2012 with 82,944processors. It currently uses 68,544 processors for a total of548,352 cores. The processors in K Computer are intercon-nected according to a special six-dimensional torus network.VLSI chips usually contain hundreds of thousands of com-ponents within an interconnection network, whilst this num-ber may reach millions in ULSI. So it is important to de-velop new families of interconnection networks with goodtopological properties for building large-scale interconnec-tion networks.
In this paper, we propose a new family of Cayley graphsWG2m
n with fixed degree, which are suitable for buildinglarge-scale interconnection networks. The remainder of thispaper is organized as follows. Section 2 gives the definitionof WG2m
n . We develop a routing algorithm and establish theupper bound of its diameter in Sect. 3. The graph-embeddingproperties and connectivity are discussed in Sects. 4 and 5,respectively. Finally, we compare WG2m
n with some othernetworks.
2 Definition of WG2mn
In this section, we define a new family of Cayley graphWG2m
n ; the graph WG2mn is isomorphic to a Cayley graph
of the wreath product Z2mwrSn, when the generator set ischosen properly.
First, we define the ranked symbol system as follows:let t1, t2, . . . , tn be n different symbols, and 2m ranks0,1,2, . . . ,2m− 1, where n ≥ 2 and m ≥ 1. We assign eachsymbol a rank, such as t ij , which means assigning rank i
to symbol tj , where 0 ≤ i ≤ 2m − 1. For n distinct sym-bols, there are n different cyclic permutations of the sym-bols. Since each symbol can be represented in 2m distinctranked forms, the vertex set of WG2m
n has a cardinality ofn · (2m)n. Let e denotes the identity permutation t0
1 t02 . . . t0
n ,
and ai11 a
i22 . . . a
inn is a string of n distinct symbols with rank
mod 2m to denote the symbolic representation of an arbi-trary vertex. Each edge of WG2m
n is of type (v, δ(v)), whereδ ∈ S = {f,f −1, g, g−1, h1, h2, . . . , hm−1} is a generatordefined as follows:
1. f (ai11 a
i22 . . . a
inn ) = a
i22 . . . a
inn a
i1+11 ;
2. f −1(ai11 a
i22 . . . a
inn ) = a
in−1n a
i11 a
i22 . . . a
in−1n−1;
Fig. 1 The Cayley graph WG42
3. g(ai11 a
i22 . . . a
inn ) = a
i22 . . . a
inn a
i11 ;
4. g−1(ai11 a
i22 . . . a
inn ) = a
inn a
i11 a
i22 . . . a
in−1n−1;
5. hj (ai11 a
i22 . . . a
inn ) = a
i11 a
i22 . . . a
in+2jn , j = 1,2, . . . ,m−1.
Figure 1 illustrates WG42. For a nonnegative integer i, we
recursively define:
f i(v) ={
v if i = 0
f • f i−1(v) if i > 2,
where • means the function composition. The notationsf −i (v), gi(v) and g−i (v) can be defined similarly. It is easyto verify f −1 •f (v) = v, g−1 •g(v) = v and hm−j •hj (v) =v. That is to say S = S−1, thus, WG2m
n are simple graphs.Since WG2
n is isomorphic to wrapped butterfly [6, 20],we only analyze the graph WG2m
n , where n ≥ 2 and m ≥ 2.Now, we prove that the constructed graph WG2m
n is Cay-ley graph. It is worth nothing that the wreath product ofthe cyclic group Z2m and symmetric group Sn,Z2mwrSn,is a group [11]. An element of the wreath product can berepresented as (g1, g2, . . . , gn;h), where h ∈ Sn and gi ∈Z2m, i = 1,2, . . . , n; the multiplication in Z2mwrSn is givenby:
(g1, g2, . . . , gn;h) · (g′1, g
′2, . . . , g
′n;h′)
= (gh′(1)g′1, . . . , gh′(n)g
′n;hh′).
Let S′={(0,0, . . . ,0;23 . . . n1), (0,0, . . . ,0;n12 . . . (n−1)),
(0,0, . . . ,0,1;23 . . . n1), (−1,0,0, . . . ,0;n12 . . . (n − 1)),
(0,0, . . . ,0,2j ;12 . . . n)}, 1 ≤ j ≤ m − 1. Then, we can de-fine the Cayley graph WG = Cay(Z2mwrSn,S
′).For a regular graph G, let V (G) and E(G) denote the
vertex set and edge set of G, respectively, and d(G) denotesthe degree of G.
Cluster Comput (2011) 14:483–490 485
Lemma 1 WG2mn is a Cayley graph when n ≥ 2, m ≥ 2.
Proof A mapping ϕ from V (WG2mn ) to V (WG) is defined
as follows:
ϕ : ai11 a
i22 . . . ain
n → (g1, g2, . . . , gn;a′1a
′2 . . . a′
n),
where gk = ik, a′i = ki iff ai = tki
.
This mapping is well-defined since any element in V (WG2mn )
has exactly one image in V (WG). For any vertex w =(g1, g2, . . . , gn;a′
1a′2 . . . a′
n) in V (WG), we can choose ai =ta′
i, ik = gk,1 ≤ i ≤ n, so the vertex a
i11 a
i22 . . . a
inn corre-
sponding to w is in V (WG2mn ). Thus, ϕ is an onto mapping.
Let u,v be adjacent vertices in WG2mn . We shall show that
their respective images in WG under ϕ are also adjacent ver-tices. For vertex u = a
i11 a
i22 · · ·ain
n , ϕ(u) = ϕ(ai11 a
i22 . . . a
inn ) =
(g1, g2, . . . , gn;a′1a
′2 . . . a′
n), because v is adjacent to u inWG2m
n , we can give the derivation as follows:
1. If v = f (u) = ai22 . . . a
inn a
i1+11 , then
ϕ(v) = ϕ(ai22 . . . ain
n ai1+11 )
= (g2, . . . , gn, g1 + 1;a′2 . . . a′
na′1)
= (g1, g2, . . . , gn;a′1a
′2 . . . a′
n)
· (0,0, . . . ,0,1;23 . . . n1)
= ϕ(u) · (0,0, . . . ,0,1;23 . . . n1),
that is to say (ϕ(u),ϕ(v)) ∈ E(WG).2. If v = f −1(u) = a
in−1n a
i11 . . . a
in−1n−1 , then
ϕ(v) = ϕ(ain−1n a
i11 . . . a
in−1n−1)
= (gn − 1, g1, . . . , gn−1;a′na
′1a
′2 . . . a′
n−1)
= (g1, g2, . . . , gn;a′1a
′2 . . . a′
n)
· (−1,0, . . . ,0;n12 . . . (n − 1))
= ϕ(u) · (−1,0, . . . ,0;n12 . . . (n − 1)),
this implies (ϕ(u),ϕ(v)) ∈ E(WG).3. If v = g(u) = a
i22 . . . a
inn a
i11 , then
ϕ(v) = ϕ(ai22 . . . ain
n ai11 )
= (g2, . . . , gn, g1;a′2 . . . a′
na′1)
= (g1, g2, . . . , gn;a′1a
′2 . . . a′
n)
· (0,0, . . . ,0;2 . . . (n − 1)n1)
= ϕ(u) · (0,0, . . . ,0;2 . . . (n − 1)n1),
that is to say (ϕ(u),ϕ(v)) ∈ E(WG).4. If v = g−1(u) = a
inn a
i11 . . . a
in−1n−1 , then
ϕ(v) = ϕ(ainn a
i11 . . . a
in−1n−1)
= (gn, g1, g2, . . . , gn−1;a′na
′1a
′2 . . . a′
n−1)
= (g1, g2, . . . , gn;a′1a
′2 . . . a′
n)
· (0,0, . . . ,0;n12 . . . (n − 1))
= ϕ(u) · (0,0, . . . ,0;n12 . . . (n − 1)),
this implies (ϕ(u),ϕ(v)) ∈ E(WG).
5. If v = hj (u) = ai11 . . . a
in−1n−1a
in+2jn , and j = 1,2, . . . ,
m − 1, then
ϕ(v) = ϕ(ai11 . . . a
in−1n−1a
in+2jn )
= (g1, g2, . . . , gn + 2j ;a′1a
′2 . . . a′
n)
= (g1, g2, . . . , gn;a′1a
′2 . . . a′
n)
· (0,0, . . . ,0,2j ;12 . . . (n − 1)n)
= ϕ(u) · (0,0, . . . ,0,2j ;12 . . . (n − 1)n),
that is to say (ϕ(u),ϕ(v)) ∈ E(WG).
Conversely, we can show if u′ and v′ are adjacent in WG,then their respective inverse images in WG2m
n are also adja-cent under ϕ−1, thus, we have WG ∼= WG2m
n . �
Theorem 1 For any n,m (n ≥ 2 and m ≥ 2), we have:
(1) d(WG2m2 ) = m + 2, and d(WG2m
n ) = m + 3 for n > 2;(2) |V (WG2m
n )| = n · (2m)n;(3) |E(WG2m
n )| = n · (2m)n · (m + 3)/2 for n > 2, and
|E(WG2m2 )| = n · (2m)n · (m + 2)/2.
Proof (1) and (2) follow from the definition of WG2mn , and
(3) follows from (1) and (2). �
3 Routing algorithm and diameter
Since WG2mn is a Cayley graph, it is vertex transitive, i.e.,
we can view the distance between any two arbitrary ver-tices as the distance between the source vertex and the iden-tity vertex by suitably renaming the symbols representingthe permutations. Thus in our subsequent discussion abouta path from a source vertex to a destination vertex, the des-tination vertex is always assumed to be the identity vertexe = t0
1 t02 . . . t0
n without loss of generality.
Definition 1 Consider an arbitrary vertex v = aj11 a
j22 . . . a
jnn
in WG2mn . Let v[i] = ji , 1 ≤ i ≤ n, denote the rank of sym-
bol ai in vertex v.
The following algorithm computes a path from an arbi-trary source vertex v = a
j11 a
j22 . . . a
jnn to the identity vertex e.
Algorithm RT(v)
1: Compute k,1 ≤ k ≤ n, such that ak = t1;2: if (k > n/2�) {3: for (1≤ i ≤ n − k + 1)
4: {if (v[n](mod 2) = 0||v[n]! = −1)
5: v = g−1(v);6: if (v[n](mod 2) = 1&&v[n]! = −1)
7: v = f −1(v); }8: else
486 Cluster Comput (2011) 14:483–490
9: for (1≤ i ≤ k − 1)
10: {if (v[1](mod 2) = 0&&v[1]! = −2)
11: v = g(v);12: if (v[1](mod 2) = 1||v[1] = −2)
13: v = f (v); } }14: for (1≤ i ≤ n)
15: {if (v[1] == −1)
16: v = f (v);17: if (v[1] == 0)
18: v = g(v);19: if (v[1](mod 2) == 0&&v[1]! = 0)
20: {v = g(v)
21: v = hm−v[n]/2(v); }22: if (v[1](mod 2) = 1&&v[1]! = −1)
23: {v = f (v);24: v = hm−v[n]/2(v); }}
Example 1 We consider the vertex u = t43 t5
4 t25 t7
1 t32 in WG8
5.The path from u to t0
1 t02 t0
3 t04 t0
5 is computed by the algorithmas follows:
t43 t5
4 t25 t7
1 t32
f −1
→ t22 t4
3 t54 t2
5 t71
g−1
→ t71 t2
2 t43 t5
4 t25
f→ t22 t4
3 t54 t2
5 t01
g→t43 t5
4 t25 t0
1 t22
h3→ t43 t5
4 t25 t0
1 t02
g→ t54 t2
5 t01 t0
2 t43
h2→ t54 t2
5 t01 t0
2 t03
f→t25 t0
1 t02 t0
3 t64
h1→ t25 t0
1 t02 t0
3 t04
g→ t01 t0
2 t03 t0
4 t25
h3→ t01 t0
2 t03 t0
4 t05 .
Theorem 2 The Algorithm RT generates a path from anyvertex a
j11 a
j22 . . . a
jnn to the identity vertex e = t0
1 t02 . . . t0
n , andthe upper bound on the diameter is �5n/2�.
Proof If k > n/2�, then n−k+1 ≤ �n/2�. After executing
lines 1–13, we obtain the vertex aj ′k
k aj ′k+1
k+1 . . . aj ′n
n aj ′
11 . . . a
j ′k−1
k−1by a path of length less than �n/2�. Then, lines 14–24 arefurther executed, we get the identity vertex e by a path oflength less than 2n. So the total number moves is at most�n/2� + 2n = �n/2� + n = �5n/2�. �
Remark The above algorithm generates a path of length≤ �5n/2� from an arbitrary vertex to the identity vertex.However, we do not know the exact diameter, while we sus-pect that the actual diameter would be of order O(n).
4 Network embeddings
Since the efficiency of many parallel algorithms is depen-dent on the topology of the interconnection network, it isuseful to explore the graph-embedding properties of WG2m
n .In this section, we show that WG2m
n can embed two types ofstructure: cycle and clique.
Definition 2 An edge introduced by the symmetric func-tions f or f −1 is called f -edge. A cycle in WG2m
n consist-ing of only f-edge is called f-cycle. Similarly, we can define
g-edge and g-cycle. The vertex v = ai11 a
i22 . . . a
inn and the ver-
tices set {hj (v)|1 ≤ j ≤ m−1} constitute a clique in WG2mn ,
which is called h-clique.
Definition 3 For each f-cycle, the vertex t01 t
i22 . . . t
inn is
called f-leader. The vertex ti11 t
i22 . . . t
inn in each g-cycle is
called g-leader. The vertex ai11 a
i22 . . . a
in−1n−1 a∗
n in each h-clique is called h-leader, where * denotes either 0 or 1. Inthe rest of this paper, we use f-leader (respectively, g-leaderand h-leader) to denote f-cycle (respectively, g-cycle andh-clique). Two cycles (cliques) are vertex-disjoint if theyhave no common vertex.
Example 2 Considering WG63, the cycle{
t01 t1
2 t23 , t1
2 t23 t1
1 , t23 t1
1 t22 , t1
1 t22 t3
3 , t22 t3
3 t21 , t3
3 t21 t3
2 , t21 t3
2 t43 , t3
2 t43 t3
1 ,
t43 t3
1 t42 , t3
1 t42 t5
3 , t42 t5
3 t41 , t5
3 t41 t5
2 , t41 t5
2 t03 , t5
2 t03 t5
1 , t03 t5
1 t02 , t5
1 t02 t1
3 ,
t02 t1
3 t01 , t1
3 t01 t1
2
}is a f-cycle with f-leader t0
1 t12 t2
3 . The cycle {t11 t1
2 t23 , t1
2 t23 t1
1 ,
t23 t1
1 t12 } is a g-cycle with g-leader t1
1 t12 t2
3 . The vertex set{t0
1 t12 t1
3 , t01 t1
2 t33 , t0
1 t12 t5
3 } is a h-clique with h-leader t01 t1
2 t13 .
Theorem 3 WG2mn can embed (2m)n−1 vertex-disjoint
f-cycles of length 2mn.
Proof Given any vertex v in WG2mn , it is easy to verify that
f 2mn(v) = v and f i(v) = f j (v), where 1≤ i, j ≤ 2mn andi = j . Thus, from an arbitrary vertex v, a f-cycle of length2mn can be generated by the functional iteration f 2mn(v).Because WG2m
n has n(2m)n vertices, totally (2m)n−1
f -cycles each of length 2mn can be generated. Thesef-cycles are vertex-disjoint by the fact that f (u) = f (v) ifand only if u = v. �
Two f-cycles F1 and F2 are said to be adjacent if thereexist vertices u ∈ F1, v ∈ F2 such that u = δ(v), where δ ∈{g,g−1, hj }. Similarly, two g-cycles G1 and G2 are adjacentif there exist vertices u ∈ G1, v ∈ G2 such that u = δ(v),where δ ∈ {f,f −1, hj }, and two h-cliques H1 and H2 areadjacent if there exist vertices u ∈ H1 and v ∈ H2 such thatu = δ(v), where δ ∈ {f,f −1, g, g−1}.
Theorem 4 Each f-cycle of WG2mn is adjacent to n(m + 1)
different f-cycles for n > 2, and any f-cycle in WG2m2 is ad-
jacent to m + 1 different f-cycles.
Proof Case 1. n > 2. For an arbitrary f-cycle with f -leaderv = t0
1 ti22 . . . t
inn ,
(1) (f −1)l • hj • f l(v) = t01 t
i22 . . . t
il+2jl . . . t
inn , for 2≤ l ≤ n
and 1≤ j ≤ m − 1;(2) (f −1)1+2nj • hj • f (v) = t0
1 ti2−2j
2 . . . tin−2jn ,
for 1≤ j ≤ m − 1;
Cluster Comput (2011) 14:483–490 487
(3) f n−1 • g(v) = t01 t
i2+12 . . . t
in+1n ;
(4) (f −1)l−1 • g−1 • f l(v) = t01 t
i22 . . . t
il+1l . . . t
inn ,
for 2≤ l ≤ n;(5) (f −1)n • g−1 • f (v) = t0
1 ti2−12 . . . t
in−1n ;
(6) (f −1)l+1 • g • f l(v) = t01 t
i22 . . . t
il+1−1l+1 . . . t
inn ,
for 2≤ l ≤ n.
According to (1)–(6), n(m + 1) leaders of n(m + 1) distinctf-cycles can be obtained from v. Therefore, each f-cycle isadjacent to n(m + 1) distinct f-cycles.
Case 2. n = 2. For an arbitrary f-cycle with f -leader v =t01 t
i22 ,
(1) (f −1)l • hj • f l(v) = t01 t
i2+2j
2 , for 1≤ j ≤ m − 1 andl(mod 2) = 0;(f −1)l+4j •hj •f l(v) = t0
1 ti2−2j
2 , for 1≤ j ≤ m−1 andl(mod 2) = 1;
(2) f • g(v) = t01 t
i2+12 ;
(3) (f −1)l−1 • g • f l(v) = t01 t
i2+12 , if l(mod 2) = 0 and 1≤
l ≤ n;(f −1)l+1 • g • f l(v) = t0
1 ti2−12 , if l(mod 2) = 1 and 1≤
l ≤ n.
Thus, each f-cycle t01 t
i22 is adjacent to m + 1 distinct f-cycles
when n = 2. �
Theorem 5 WG2mn can embed (2m)n vertex-disjoint g-cy-
lces of length n.
Proof Similar to Theorem 3. �
Theorem 6 Each g-cycle of WG2mn is adjacent to n(m + 1)
different g-cycles.
Proof Similar to Theorem 4, for an arbitrary g-cycle withg-leader v = t
i11 t
i22 . . . t
inn , we have
(1) (g−1)l • hj • gl(v) = ti11 t
i22 . . . t
il+2jl . . . t
inn ,
for 1≤ j ≤ m − 1;(2) (g−1)l+1 • f • gl(v) = t
i11 t
i22 . . . t
il+1l . . . t
inn ,
for 0≤ l ≤ n − 1;(3) (g−1)l−1 • f −1 • gl(v) = t
i11 t
i22 . . . t
il−1l . . . t
inn ,
for 1≤ l ≤ n.
According to (1)–(3), g-cycle ti11 t
i22 . . . t
inn is adjacent to
n(m + 1) distinct g-cycles. �
Theorem 7 WG2mn can embed n(2m)n/m vertex-disjoint h-
clique of size m.
Proof For an arbitrary vertex v = ti11 t
i22 . . . t
inn , the vertex
set H = {v} ∪ {hj (v)|1 ≤ j ≤ m − 1} constitute a h-clique,where |H | = m. So, WG2m
n can embed n(2m)n/m vertex-disjoint h-clique of size m. �
Theorem 8 Each h-clique of WG2mn is adjacent to 4m dif-
ferent h-cliques for n > 2, and any h-clique of WG2m2 is ad-
jacent to 3m different h-cliques.
Proof Case 1. n > 2. Given an arbitrary h-clique H withh-leader v = a
i11 a
i22 . . . a
in−1n−1 a∗
n , we can get the following re-sults:
(1) v1 = f (v) = ai22 . . . a
in−1n−1 a∗
nai1+11 ∈{
ai22 . . . a
in−1n−1a∗
na01 if i1(mod 2) = 1;
ai22 . . . a
in−1n−1a∗
na11 if i1(mod 2) = 0.
(2) v2 = g(v) = ai22 . . . a
in−1n−1 a∗
nai11 ∈{
ai22 . . . a
in−1n−1a∗
na01 if i1(mod 2) = 0;
ai22 . . . a
in−1n−1a∗
na11 if i1(mod 2) = 1.
(3) v3 = f −1(v) = a∗−1n a
i11 a
i22 . . . a
in−1n−1 ∈{
a∗−1n a
i11 a
i22 . . . a0
n−1 if in−1(mod 2) = 0;a∗−1n a
i11 a
i22 . . . a1
n−1 if in−1(mod 2) = 1.
(4) v4 = g−1(v) = a∗na
i11 a
i22 . . . a
in−1n−1 ∈{
a∗na
i11 a
i22 . . . a0
n−1 if in−1(mod 2) = 0;a∗na
i11 a
i22 . . . a1
n−1 if in−1(mod 2) = 1.
Four distinct h-cliques can be obtained from v. Since|V (H)| = m, each h-clique of WG2m
n is adjacent to 4m dif-ferent h-cliques.
Case 2. n = 2. Similar to Case 1. �
5 Maximal fault tolerance
The fault tolerance of a graph is measured by the connec-tivity of the graph [21]. The connectity ξ(G) of a graphG is the minimum size of a vertex set S such that G–S isdisconnected or only one vertex. A graph G is said to beκ-connected if its connectivity is at least κ . Obviously, theconnectivity of G cannot exceed the minimum degree ofa vertex in G; thus, ξ(WG2m
n ) ≤ m + 3 when n > 2, andξ(WG2m
2 ) ≤ m − 2. A graph is said to have maximal faulttolerance if its connectivity equals the minimum degree ofthe given graph. In this section, we show that WG2m
n aremaximal fault tolerant.
Definition 4 For a given WG2mn , the reduced graph RWG2m
n
of WG2mn can be constructed using the following two steps:
(1) For each f-cycle t01 t
i22 . . . t
inn , shrink it to a single vertex,
and label it with ti22 . . . t
inn .
(2) Connect two vertices ti22 . . . t
inn and t
i′22 . . . t
i′nn in RWG2m
n ,iff there is only one ik who satisfies that ik − i′k = ±1 or2j , where 1≤ j ≤ m − 1 and 2≤ k ≤ n.
488 Cluster Comput (2011) 14:483–490
It is obviously that V (RWG2mn ) = (2m)n−1 and d(RWG2m
n )
= (n − 1)(m + 1).
Lemma 2 ξ(RWG2mn ) = (n − 1)(m + 1).
Proof Case 1. n = 2. RWG2m2 is a chordal ring, and
ξ(RWG2m2 ) = m + 1.
Case 2. n ≥ 3. It is easy to verify RWG2mn = C1�C2�
· · ·�Cn−1, where Ci = RWG2m2 for 1≤ i ≤ n−1, and � de-
notes Cartesian product. According to Proposition 9 in [8],we have
ξ(RWG2mn ) = ξ(C1) + ξ(C2) + · · · + ξ(Cn−1)
= (n − 1)(m + 1). �
We need the following lemma to prove our main result,and the proof of the lemma can be found in [9].
Lemma 3 Given two sets of vertices V1 and V2 such that|V1| = |V2| = μ in a μ-connected graph, there are μ vertex-disjoint paths connecting the vertices from V1 to V2.
Theorem 9 ξ(WG2m2 ) = m + 2, and ξ(WG2m
n ) = m + 3 forn ≥ 3.
Proof For any two vertices u and v in WG2mn , we construct
maximal vertex-disjoint paths between them. There are twocases including n = 2 and n ≥ 3.
Case 1. n = 2. Assume u ∈ t01 t
i22 , then g(u) ∈ t0
1 ti2±12
and hj (u) ∈ t01 t
i2±2j
2 . We only consider g(u) ∈ t01 t
i2+12 and
hj (u) ∈ t01 t
i2+2j
2 for simplicity.
Case 1.1. u and v are in the same f-cycle t01 t
i22 , let v = f l(u).
The cycle t01 t
i22 is a concatenation of two internally disjoint
paths form u to v, which contributes two paths. The other m
disjoint paths can be constructed as following two steps:
(1) Let u′ = hj (u),1 ≤ j ≤ m − 1, then u′ ∈ t01 t
i2+2j
2 .
• If l(mod 2) = 0, then v = hm−j • f l • hj (u).• If l(mod 2) = 1, then v = hm−j • (f −1)4j • f l •
hj (u).
Then, a path can be constructed from u to v. Thus, wecan construct m − 1 vertex-disjoint paths by using the set{hj |1 ≤ j ≤ m − 1}.(2) Let u′ = g(u), then u′ ∈ t0
1 ti2+12 .
• If l(mod 2) = 0, then v = g−1 • f l • g(u).• If l(mod 2) = 0, then v = g−1 • hm−1 • f l+2 • g(u).
Then, the last path from u to v can be constructed.Case 1.2. u and v are in the different f-cycles. Assume u ∈t01 t
i22 , v ∈ t0
1 ti′22 , while t0
1 ti22 = t0
1 ti′22 .
Case 1.2.1. u and v are adjacent, then v = g(u) or v =hj (u),1 ≤ j ≤ m − 1.
(1) If v = g(u), then t01 t
i′22 = t0
1 ti2+12 .
The vertex-disjoint paths from u to v can be constructed asfollow two steps:
(1.1) It is clear that v = g(u), v = f 2 • g • (f −1)2(u) andv = (f −1)2 • g • f 2(u). Then, we can construct threedisjoint paths.
(1.2) Let u′ = hj (u), then u′ ∈ t01 t
i2+2j
2 ,1 ≤ j ≤ m − 1.
Since v = hm+j • g • (f −1)4j • hj (u), a path can be con-structed. Thus, we can construct m − 1 vertex-disjoint pathsfrom u to v by using the set {hj |1 ≤ j ≤ m − 1}.(2) If v = hj (u), where 1≤ j ≤ m−1, then t0
1 ti′22 = t0
1 ti2+2j
2 .
The vertex-disjoint paths from u to v can be constructed asfollow three steps:
(2.1) Since v = hj (u), v = f 2 • hj • (f −1)2(u) and v =(f −1)2 • hj • f 2(u). Then, we can construct threepaths.
(2.2) Since v = hm−j ′+j • hj ′(u), where j ′ = j , we canconstruct m − 2 paths by using the set {hj ′ |1 ≤ j ′ ≤m − 1, j ′ = j}.
(2.3) Since v = g • hm−j • f 4j • g(u), the last path can beconstructed.
Case 1.2.2. t01 t
i22 is adjacent to t0
1 ti′22 , while u and v are
not adjacent. According to Theorem 4, t01 t
i22 is adjacent to
m+ 1 different f-cycles. Assume t01 t
i′22 = t0
1 ti2−12 , the desired
vertex-disjoint paths from u to v can be constructed as fol-low two steps:
(1) Let v1 = g • f −1(u), v2 = g • f (u), then v1 and v2 be-long to t0
1 ti2−12 . The two disjoint paths from v1 to v and
v2 to v can be constructed along t01 t
i2−12 . Then we can
construct two vertex-disjoint paths from u to v.(2) The set of vertices {g(u),h1(u), . . . , hm−1(u)} (re-
spectively, {g(v),h1(v), . . . , hm−1(v)}) are in m dif-ferent f-cycles denoted by {C1,C2, . . . ,Cm} (respec-tively, {C′
1,C′2, . . . ,C
′m}), can be reached from u (re-
spectively, v). By Lemma 2 and Lemma 3, there are m
vertex-disjoint paths connecting {C1,C2, . . . ,Cm} and{C′
1,C′2, . . . ,C
′m} in RWG2m
2 . This implies that thereexist k pairwise internally disjoint paths from u to v inWG2m
2 .
Case 1.2.3. t01 t
i22 is not adjacent to t0
1 ti′22 . Assume t0
1 ti′22 =
t01 t
i2+2j+12 , then t0
1 ti2+12 is adjacent to both t0
1 ti22 and t0
1 ti2+2j+12 .
Consider the following four cases:
(a) If g(u) ∈ t01 t
i2+12 , then vertex g • f 2(u) belongs to
t01 t
i2+12 . Two paths can be constructed from u to t0
1 ti2+12 .
The vertices set {g • f −1(u),h1(u), . . . , hm−1(u)} canbe constructed, which are in m different f-cycles denotedby Cu1.
Cluster Comput (2011) 14:483–490 489
(b) If g(u) /∈ t01 t
i2+12 , then vertices g • f −1(u) and g • f (u)
belong to t01 t
i2+12 .Then, we can construct two paths from
u to t01 t
i2+12 . The vertices set {g(u),h1(u), . . . , hm−1(u)}
can be constructed, which are in m different f-cycles de-noted by Cu2.
(c) If hm−j (v) ∈ t01 t
i2+12 , then vertex hm−j • f 2(v) belongs
to t01 t
i2+12 . Two paths can be constructed from v′
1 to
t01 t
i2+12 . The vertices set {g(v), g • f −1(v), h1(v), . . . ,
hm−j−1(v), hm−j+1(v), . . . , hm−1(v)} can be construct-ed, which are in m different f-cycles denoted by Cv1.
(d) If hm−j (v) /∈ t01 t
i2+12 , then vertices hm−j • f −1(v) and
hm−j • f (v) belong to t01 t
i2+12 . We can construct two
path from v to t01 t
i2+12 . The vertex set {g(v),h1(v),
. . . , hm−1(v)}, which are in m different f-cycles denotedby Cv2.
According to (a)–(d), we can construct two vertex-disjoint paths from u to v through t0
1 ti2+12 . Similar to step 2
in Case 1.2.2, we can construct m disjoint paths by usingf-cycle set Cu1 (or Cu2) and Cv1 (or Cv2).
Case 2. n ≥ 3.Case 2.1. u and v are in the same f-cycle t0
1 ti22 . . . t
inn . Assume
v = f l(u), then there are two vertex-disjoint paths from u tov along t0
1 ti22 . . . t
inn . The other m+1 paths can be constructed
as follows:
1. For each generator hj , where 1≤ j ≤ m − 1, we havev = hm−j • f l • hm−j • (f −1)l • hj • f l • hj (u). Then,we can construct m − 1 vertex-disjoint paths from u to v
by using the set {hj ′ |1 ≤ j ≤ m − 1}.2. Since v = g • (f −1)n−l+1 • gn−l • f • gl−2 • f n • g(u),
we can construct a vertex-disjoint path.3. For generator g−1, we can construct a path similar to 2
of Case 2.1.
Case 2.2. u and v are in the different f-cycles. The setof vertices {g • f (u), g • f −1(u), g(u), g−1(u),h1(u), . . . ,
hm−1(u)} (respectively, {g • f (v), g • f −1(v),
g(v), g−1(v), h1(v), . . . , hm−1(v)}) are in m + 3 differ-ent f-cycles denoted by {C1,C2, . . . ,Cm+3} (respectively,{C′
1,C′2, . . . ,C
′m+3}), can be reached from u (respectively,
v). Similar to step 2 in Case 1.2.2, we can construct m + 3vertex-disjoint paths from u to v in WG2m
n . �
6 Conclusion
In this section, we consider some important parameters,such as vertex degree (d), diameter (D) and their product(d × D), namely cost measure. From Table 1, we can seethat, when WG2m
n and Gk,n have the same number of ver-tices and approximate diameter the degree of WG2m
n is muchsmaller.
Table 1 Comparison of degree and diameter between WG2mn and Gk,n
Network No. nodes Degree Diameter
WG65 5 · 65 6 12
WG85 5 · 85 7 12
WG105 5 · 105 8 12
WG125 5 · 125 9 12
WG145 5 · 145 10 12
WG2mn n(2m)n m + 3 or m + 2 ≤ �5n/2�
G7,5 5 · 65 7 10
G9,5 5 · 85 9 10
G11,5 5 · 105 11 10
G13,5 5 · 125 13 10
G15,5 5 · 145 15 10
Gk,n n(k − 1)n k ≤ �5n/2� − 2 or 2n
Fig. 2 Relationship between dD and node number
In Fig. 2, we compare the cost measure of WG2mn
with that of the other three structures: Hypercube(Hn),Star graph(Sn) and Gk,n. These three structures are allbased on the wreath product ZmwrSn, where Zm is a cyclicgroup of order m and Sn is a symmetric group. The Hyper-cube and Star graph have been widely used to construct realnetwork architecture, while Gk,n has not been used in anyreal application by now.
We can see, with the increase of the scale (number ofvertices) of the networks, the cost measure of Hypercube in-creases dramatically, the quantities of Sn,Gk,n and WG2m
n
are similar at first, and then the cost measure of WG2mn in-
creases much slowly. This means WG2mn is more suitable to
design large-scale network structures than Sn and Gk,n.However, the optimal routing algorithm and exact diame-
ter of WG2mn are not known. The basic communication mod-
els and parallel applications based on this proposed modelwould be developed in the future.
490 Cluster Comput (2011) 14:483–490
Acknowledgements The paper was supported by Guangdong Natu-ral Science Foundation Item (S2011040003481). The authors are grate-ful to the anonymous referees for valuable suggestions.
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Zhen Zhang received the Ph.D.Degree in Computer Science fromSouth China University of Tech-nology, Peoples Republic of China,in 2011. He is currently a lecturein Department of Computer Sci-ence, Jinan University, Guangzhou,People’s Republic of China. Hisresearch interests include discretemathematics, parallel and distributedcomputing, complex networks.
Wenjun Xiao received the Ph.D.degree in Mathematics from SichuanUniversity, People’s Republic ofChina, in 1989. Currently, he is pro-fessor in the School of ComputerScience and Engineering, SouthChina University of Technology,Guangzhou, People’s Republic ofChina. His research interests in-clude discrete mathematics, paral-lel and distributed computing, com-plex networks, and software archi-tecture. He has published more than60 papers in international confer-ences and journals, including IEEE
Trans. Computers and IEEE Trans. Parallel and Distributed Systemson these topics since 1985.
