Reliability measures in relation to the h-extra edge-connectivity of folded hypercubes

Theoretical Computer Science 615 (2016) 71–77

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Reliability measures in relation to the h-extra edge-connectivity of folded hypercubes ✩ Mingzu Zhang, Lianzhu Zhang ∗ , Xing Feng School of Mathematical Science, Xiamen University, Fujian 361005, China

a r t i c l e

i n f o

Article history: Received 12 October 2015 Accepted 29 November 2015 Available online 9 December 2015 Communicated by S.-y. Hsieh Keywords: Fault tolerance Extra edge-connectivity Folded hypercube Multiprocessor

a b s t r a c t The folded hypercube FQ n , as a variation of the hypercube Q n , was proposed by A. El-Amawy and S. Latifi in 1991. The h-extra edge-connectivity of the underlying topological graph of a multiprocessor system is a kind of measure for the reliability of the multiprocessor system. In this paper, we determine the exact value of λh (FQ n ) for n integer h, 1 ≤ h ≤ 2 2 +1 and 6 ≤ n, which generalizes several known results for h ≤ n. n More interestingly, we also show that λh (FQ n ) is the constant ( n2  − r + 1)2 2 +r for n

n

2 2 +r − lr ≤ h ≤ 2 2 +r , where r = 1, 2, . . . ,  n2  − 1 and lr = lr =

2r +1

2

3

−2

if n is even. In particular, for r =

 n2  − 1,

 2 3+2  n

22r −1 3 n−1

if n is odd and

≤ h ≤ 2 , λh (FQ n ) = 2n . © 2015 Elsevier B.V. All rights reserved.

1. Introduction With the rapid development of VLSI technology, a multiprocessor system may contain a tremendous number of nodes and links. The underlying topology of a multiprocessor system is often modeled by a connected graph in which the vertices represent processors and the edges represent communication links between processors. Once a system is implemented, some of processors or links may be faulty. Designing such a system without defects is impossible. The Menger’s edge connectivity of multiprocessor systems is one of the most famous types of factors to measure the edge fault tolerance. However, the classical edge-connectivity λ(G ), as a parameter for measuring the fault tolerance of network systems with few processors, is not suitable for large-scale processing systems. It is almost impossible for all links incident to the same processor to fall simultaneously. So it usually underestimates the resilience of large networks [13,22,25,26]. To compensate for this shortcoming, several new concepts on the edge-connectivity of graphs were proposed which can be called conditional connectivity by Harary [8]. One of them is the extra edge-connectivity. Extra edge connectivity was introduced by Fàbrega and Foil [7]. For a given positive integer h, an h-extra edge-cut of a connected graph G is defined as an edge set F of graph G, whose deletion yields a disconnected graph with all its components having at least h vertices. The h-extra edge-connectivity of a connected graph G, denoted by λh (G ), is the minimum cardinality taken over all h-extra edge-cuts of G. It is obvious that λ1 (G ) = λ(G ). Recently, the h-extra edge connectivity of particular classes of graphs has received extensive attention [2,4,6–9,13,16,19,21–26] and so on. This paper focuses on determining the exact values of h-extra edge-connectivity of the folded hypercube.

✩

*

The research is supported by National Natural Science Foundation of China (Grant Nos. 11171279 and 11471273). Corresponding author. E-mail addresses: [email protected] (M. Zhang), [email protected] (L. Zhang), [email protected] (X. Feng).

http://dx.doi.org/10.1016/j.tcs.2015.11.049 0304-3975/© 2015 Elsevier B.V. All rights reserved.

72

M. Zhang et al. / Theoretical Computer Science 615 (2016) 71–77

Table 1 Results on λh (FQ n ). h

λh (FQ n )

Authors

1 2 3 4

n + 1, n ≥ 2 2n, n ≥ 2 3n − 1, n ≥ 5 4n − 4, n ≥ 5 ξh (FQ n ), n ≥ 6

A. El-Amawy and S. Latifi [5] Q. Zhu and J. Xu [23] Q. Zhu, J. Xu, X. Hou and M. Xu [24] N.-W. Chang, C.-Y. Tsai and S.-Y. Hsieh [4] W. Yang and H. Li [22]

ξh (FQ n ), n ≥ 4

Current authors

≤n n+1 ≤ 2 2 +1 − 4, for odd n n ≤ 2 2 +1 − 2, for even n n+1 n+1 2 2 +1 − 3, 2 2 +1 − 2, for odd n n−1 n−1 n−1 + r + , 2 2 r − 1, . . . , 2 2 +r 2 2 22r +1 −2 − 3 , for odd n, r = 1, 2 . . . , n+2 1 − 1 n n n 2r +1 2 2 +r , 2 2 +r − 1, . . . , 2 2 +r − 2 3 −2 , for even n, r = 1, 2 . . . , n2 − 1

n

 n2 2 2 +1 , n ≥ 4 n ( n2  − r + 1)2 2 +r , n ≥ 3

Current authors Current authors

As a variant of the hypercube, the folded hypercube first proposed by El-Amawy and Latifi [5], is one of the most potential interconnected networks [18,20]. Compared to an n-dimensional hypercube, the n-dimensional folded hypercube with the same number of nodes has larger vertex degree, highly symmetric property, vertex-transitivity and half the diameter. In parallel computing, they have been used as underlying topologies of several parallel systems, such as ATM switches [14,15], PM2I networks [11,12], and 3D-FolH-NOC networks. Definition 1.1. For an integer n ≥ 1, the n-dimensional hypercube, denoted by Q n , is a connected graph with 2n vertices. The vertex set V ( Q n ) = {xn xn−1 . . . x1 : xi ∈ {0, 1}, 1 ≤ i ≤ n} is the set of all n-bit binary strings. Two vertices u = un un−1 . . . u 1 and v = v n v n−1 . . . v 1 of Q n are adjacent if and only if they differ in exact one position. Definition 1.2. Two vertices x = xn xn−1 . . . x1 and y = yn yn−1 . . . y 1 of Q n are complementary if and only if the bits of x and y are complements of each other, that is, y i = xi = 1 − xi for each i = 1, 2, . . . , n. The n-dimensional folded hypercube, denoted by FQ n , is an undirected graph obtained from Q n by adding extra 2n−1 edges with two vertices of every edge complementary each other. These edges are called complementary edges, denoted by M. It has been shown that Q n is n-regular n-connected and FQ n is (n + 1)-regular (n + 1)-connected [5]. Several authors have strived to crack the nut of determining the h-extra edge-connectivity of folded hypercubes these years. They gave the exact values of λh (FQ n ) for some special cases 1 ≤ h ≤ 4 [4,5,23,24]. Instead of considering the results for some special cases, very recently Yang and Li [22] made an effort to obtain a general results for each h ≤ n, 6 ≤ n. Since for any integer m, it has a unique binary representation, for 1 ≤ m ≤ 2n−1 , it can be written that m = t i = log2 (m −

i −1 r =0

s 

i =0

2t i , where t 0 = log2 m,

2tr ), and t 0 > t 1 > · · · > t s . Let

ξm (FQ n ) = (n + 1)m −

s 

t i 2t i +

i =0

s 

2 · i · 2t i .

(1)

i =0

Motivated by known results, we go further and focus on the cases for n < h ≤ 2n−1 . In this paper, we discuss the monotonicity of function ξm (FQ n ) and some other properties for 1 ≤ m ≤ 2n−1 . As an application, we determine the value n of λh (FQ n ) for 1 ≤ h ≤ 2 2 +1 , n ≥ 4. This result generalizes all of previous results for h ≤ n. Furthermore, it is shown that n

n

n

λh (FQ n ) is a constant ( n2  − r + 1)2 2 +r for 2 2 +r − lr ≤ h ≤ 2 2 +r , where r = 1, 2, . . . ,  n2  − 1 and lr = 2r +1

22r −1 3

if n is

odd and lr = 2 3 −2 if n is even. In particular, for r =  n2  − 1,  2 3+2  ≤ h ≤ 2n−1 , λh (FQ n ) = 2n . Not only do these results generalize the previous results on the h-extra edge-connectivity of FQ n , but also obtain conclusions for many new cases. For the sake of convenience, we summarize those results in Table 1. n

2. Preliminaries Recall the definition of the h-extra edge-connectivity of a connected graph G, λh (G ), is the minimum number of an edge set of graph G whose removal disconnects the graph G with all its components having at least h vertices. Given a vertex set X ⊂ V (G ), we denote the set of edges of G in which each edge contains exactly one end in X and the other in X = V (G ) \ X by [ X , X ]. Since the optimal number is obtained only when there are exactly two components produced in G − [ X , X ]. Let

ζm (G ) = min{|[ X , X ]| : | X | = m ≤ 

| V (G )| 2

, and both G [ X ] and G [ X ] are connected}.

M. Zhang et al. / Theoretical Computer Science 615 (2016) 71–77

Thus, for each 1 ≤ h ≤ 

| V (G )| 2

73

,

λh (G ) = min{ζm (G ) : h ≤ m ≤ 

| V (G )| 2

}.

(2)

For an d-regular graph, one can easily obtained that

ζm (G ) = dm − exm (G ),

(3)

where exm (G ) is twice of the maximum number edges of the subgraph of G induced by m vertices. The several results focused on the induced subgraph (of Q n and FQ n ) with maximum number of edges [1,3,10,13,17]. Definition 2.1. (See [1].) A set of m vertices of Q n (resp. FQ n ) is said to be a composite set for m if the number of edges of the subgraph induced by these m vertices is not less than the number of edges of subgraphs induced by any other set of m vertices of Q n (resp. FQ n ). A composite hypercube of Q n (resp. composite folded hypercube of FQ n ) is defined to be a subgraph of Q n (resp. FQ n ), which is induced by some composite set of Q n (resp. FQ n ). For convenience, the vertex x = xn xn−1 . . . x1 of n-dimensional hypercube and n-dimensional folded hypercube also can be represented by decimal number

n 

i =1

xi 2i −1 in this paper.

Definition 2.2. (See [10,17].) The subgraph induced by vertex set {0, 1, . . . , m − 1} (under decimal representation) of Q n (resp. FQ n ), denoted by L m (resp. LF m ), is called as an incomplete hypercube (resp. incomplete folded hypercube) on m vertices of Q n (resp. FQ n ). Definition 2.3. (See [17].) A reverse incomplete hypercube (resp. reverse incomplete folded hypercube) on m vertices of Q n (resp. FQ n ) is the subgraph induced by {2n − 1, 2n − 2, . . . , 2n − m} and is denoted by R m (resp. RF m ), for 1 ≤ m ≤ 2n . Lemma 2.4. (See [17].) L m is isomorphic to R m , and LF m isomorphic to RF m , for 1 ≤ m ≤ 2n . If we delete the edges [ V (LF m ), V (LF m )], both LF m and RF 2n −m are connected. Lemma 2.5. (See [3,10,17].) For 1 ≤ m ≤ 2n , both V ( L m ) and V ( R m ), (resp. V (LF m ) and V (RF m )) are composite sets of Q n (resp. FQ n ). 3. Properties of the function ξm (FQ n ) For Q n and FQ n and a positive integer m = [13,17,22], one can rewrite that

2| E (LF m )| =

s  i =0

2t i ≤ 2n , combining the results discussed above with the conclusions in

⎧ s s   ⎪ ⎪ ⎪ t i 2t i + 2i2t i ⎪ ⎨ i =0

i =0

i =0

i =0

if 1 ≤ m ≤ 2n−1 ; (4)

s s   ⎪ ⎪ ti ⎪ t 2 + 2i2t i + m − 2n−1 if 2n−1 < m ≤ 2n . ⎪ i ⎩

By Lemma 2.5, V (LF m ) is a composite set of folded hypercubes FQ n . Hence exm (FQ n ) = 2| E (LF m )|. Since folded hypercube FQ n is (n + 1)-regular and by application of the formulas (1), (3) and (4), we have, for 1 ≤ m =

ξm (FQ n ) = ζm (FQ n ) = (n + 1)m −

s  i =0

t i 2t i −

s 

s 

i =0

2t i ≤ 2n−1 ,

2i2t i .

i =0

For m ≤ 2n−1 , note that exm (FQ n ) and exm ( Q n ) are interchangeable each other, if no confusion arises. In the following, we prove some properties of the function ξm (FQ n ) in terms of m. Lemma 3.1. Let m =

s −1 i =0

2t i ≤ 2n−1 − 2. Then

ξm+1 (FQ n ) − ξm (FQ n ) = (n + 1) − 2s.

(5)

74

M. Zhang et al. / Theoretical Computer Science 615 (2016) 71–77

Proof. Because of m ≤ 2n−1 − 2, no complementary edges is contained in LF m+1 . Since V (LF m+1 ) = {m} ∪ V (LF m ) =

{0, 1, 2, . . . , m − 1, m}, m =

s −1

−1  s

2t i >

i =0

i =0



2t i −2t i , i = 0, 1, . . . , s − 1, and m <

−1 s i =0



2t i + 2e , n − 1 ≥ e = t i , i = 0, 1, . . . , s − 1,

the neighbors of vertex m in LF m+1 are exactly obtained from the binary representation m by changing only one position’s

1 to 0 . There are exactly s edges joining m

s −1

=

2t i to s vertices

−1 s

i =0

i =0



2t i − 2t i , i = 0, 1, . . . , s − 1 in incomplete folded

hypercube LF m+1 . Hence, by formulas (3), (4) and (5), one can deduce that

ξm+1 (FQ n ) − ξm (FQ n ) = [(n + 1)(m + 1) − exm+1 (FQ n )] − [(n + 1)m − exm (FQ n )] = (n + 1) + (exm (FQ n ) − exm+1 (FQ n )) = (n + 1) + (2| E (FLm )| − 2| E (FLm+1 )|) = (n + 1) − 2s.

2 n

Corollary 3.2. The function ξm (FQ n ) of integer m in the interval [1, 2 2 +1 − 1] is monotonous increasing. n

Proof. It is enough to prove that ξm+1 (FQ n ) − ξm (FQ n ) ≥ 0 for 1 ≤ m ≤ 2 2 +1 − 2. Suppose that m = n

since m ≤ 2 2 +1 − 2. Thus, by Lemma 3.1,

n

ξm+1 (FQ n ) − ξm (FQ n ) = (n + 1) − 2s ≥ (n + 1) − 2  ≥ 0.

s −1 i =0

2t i . Then s ≤  n2 

2

2

n n Lemma 3.3. Suppose 2 2 +1 − dr < m < 2 2 +1 , where dr = 4 if n is odd and dr = 2 if n is even. Then ξm (FQ n ) > ξ

ξ

 n +1 2 2 −dr

(FQ n ) =

2

n  n2 2 2 +1 .

Proof. We show ξ

2

n

n +1 2

n

(FQ n ) = (n + 1)2 2 +1 − ex

2

n +1 2

2

n +1 2

(FQ n ) =

n

(FQ n ) = (n + 1)2 2 +1 − ( n2  + 1)2 2 +1 =  n2 2 2 +1 . By Corol n2 +1

lary 3.2, we have ξm+1 (FQ n ) − ξm (FQ n ) ≥ 0 for each 1 ≤ m ≤ 2

ξ

n

n +1 2

− 2. It is sufficient to show that ξ

2

(FQ n ).

n +1 2

−dr

(FQ n ) =

If n is odd, by calculating, we can obtain

ξ

2

n +1 +1 2 −4

(FQ n ) = (n + 1)(2 = (n + 1)(2 = = = =

n−1 2

n−1 2

n−1 2

n−1 2

n +1 +1 2

− 22 ) − ex

2

n +1 +1 2

2

(2

n +1 +1 2

(2

n +1 2 +1

2

n +1 2 +1

2

n +1 +1 2

− 22 ) − ex

− 22 ) + ( − 22 ) + (

+(

n+1 2

n +1 +1 2 −2 2

(FQ n )

n +1 +1 2 −2 2

(FQ

n+1 2

n+1

−

+ 1)(2

n +1 +1 2

)

− 22 ) − ex

2

2

n +1 +1 2

n +1 +1 2 −2 2

(FQ

2

n +1 +1 2

)

)

+ 1)22 − 2 × 22

.

One can easily check that the result holds for even n, since ξ proof is done.

n +1 +1 2

+ 1)22 − ex22 (FQ

2 n−1 2

2

2

n

n +1

22

−2

(FQ n ) = n2 2 2 +2 + ( n2 −

n 2

Lemma 3.4. Let 2c < m ≤ 2n−1 for 0 ≤ c ≤ n − 2. Then

ξm (FQ n ) ≥ ξ2c (FQ n ). Proof. It is enough to prove that for 2k < m ≤ 2k+1 , k = c , c + 1, . . . , n − 2, ξm (FQ n ) ≥ ξ2k (FQ n ), since

ξ2k+1 (FQ n ) − ξ2k (FQ n ) = (n + 1)2k+1 − ex2k+1 (FQ n ) − (n + 1)2k − ex2k (FQ n ) = [(n + 1)2k+1 − (k + 1)2k+1 ] − [(n + 1)2k − k2k ] = (n + 1 − k − 1)2k+1 − (n + 1 − k)2k = (n + 1 − k − 2)2k > 0.

n

+ 1)21 − 1 × 21 = n2 2 2 +1 . The

M. Zhang et al. / Theoretical Computer Science 615 (2016) 71–77

Therefore, we may assume that 2k < m < 2k+1 . So 0 < m − 2k < 2k . Let m = t 1 < k ≤ n − 2. Then m =

s  i =1

2t i =

s −1 i =0

s  i =0

75

2t i and m = m − 2k . Clearly, t 0 = k and

2t i+1 < 2k ≤ 2n−2 and

ξm (FQ n ) − ξ2k (FQ n ) = [(n + 1)m − exm (FQ n )] − [(n + 1)2k − ex2k (FQ n )] s s s    2t i − t i 2t i − 2i2t i − (n + 1)2k + k2k = (n + 1) i =0

= (n + 1)

s −1 

i =0

2t i+1 −

i =0

s −1 

i =0

t i +1 2t i+1 −

i =0

= (n − 1)m −

s −1 

s −1 

2(i + 1)2t i+1

i =0

t i +1 2t i+1 −

s −1 

i =0

2i2t i+1 .

i =0

For 0 < m < 2k , using the formula (4), we have exm (FQ n ) < km and ξm (FQ n ) − ξ2k (FQ n ) = (n − 1)m − s −1 i =0

2i2t i+1 > (n − 1 − k)m > 0. n

2

 n2 +r

i =0

t i +1 2t i+1 −

2 n

Lemma 3.5. Suppose 2 2 +r − lr ≤ m ≤ 2 2 +r , where r = 1, 2, . . . ,  n2  − 1 and lr = Then

ξm (FQ n ) ≥ ξ

s −1

22r −1 3

if n is odd and lr =

22r +1 −2 3

if n is even.

(FQ n ).

Proof. Let f = 0 if n is odd, and f = 1 if n is even. For r = 1, 2, . . . ,  n2  − 1 and 0 ≤ kr ≤ r, we define M r ,kr as follows:

M r ,kr =

⎧ n  2 +r ⎪ ⎪ ⎨2 n

⎪ 2 2 +r − ⎪ ⎩

if kr = 0; k r −1

22r −2i −2+ f

if 1 ≤ kr ≤ r .

i =0 n

One can check that M r ,kr − M r ,kr +1 = 22r −2kr + f −2 , for any 0 ≤ kr ≤ r − 1, and M r ,r = 2 2 +r −  n2 +r

22r + f −2 f 3

n . For 2 2 +r − lr ≤ m ≤

2 , there is a positive integer m such that 0 ≤ m ≤ 22r −2kr + f −2 and m = M r ,kr − m . Owing to exm (FQ n ) = exm ( Q n ) for m ≤ 2n−1 , one can obtained that

n

n

2 n

2 n

2 n

2 n

2 n

2

(n + 1)m − exm (FQ n ) = (  − r + 1)m + (  + r )m − exm ( Q n ) = (  − r + 1)m + (  + r )m − exm ( Q  n +r ) 2

= (  − r + 1)m + (  + r )(2 = (  − r + 1)2 2 − ex

2

 n2 +r

−m

 n2 +r

 n2 +r

− m) − ex

2

n

− (  − r + 1)(2

 n2 +r

2

 n2 +r

−m

( Q  n +r ) 2

n

n

− m) + (  + r )(2 2 +r − m) 2

( Q  n +r ) 2

n

= ξ Mr,0 (FQ n ) + (2r − 2 + f )(2 2 +r − m) − ex

2

 n2 +r

−m

( Q  n +r ), 2

and

ex

2

n +r 2

−m

( Q  n +r ) = ex

2

2

=

 n2 +r

− M r ,kr

( Q  n +r ) + exm ( Q  n +r ) + 2kr m 2

2

k r −1

k r −1

i =0

i =0

(2r − 2i − 2 + f )22r −2i −2+ f +

= (2r − 2 + f )

k r −1 i =0

2i22r −2i −2+ f + exm ( Q  n +r ) + 2kr m 2

22r −2i −2+ f + exm ( Q  n +r ) + 2kr m 2

n

= (2r − 2 + f )(2 2 +r − M r ,kr ) + exm ( Q  n +r ) + 2kr m , 2

therefore,

76

M. Zhang et al. / Theoretical Computer Science 615 (2016) 71–77

ξm (FQ n ) = (n + 1)m − exm (FQ n ) = ξ Mr,0 (FQ n ) + (2r − 2 + f )( M r ,kr − m) − exm ( Q  n +r ) − 2kr m 2

= ξ Mr,0 (FQ n ) + (2r − 2kr + f − 2)m − exm ( Q 2r −2kr + f −2 ) = ξ Mr,0 (FQ n ) + ξm ( Q 2r −2kr + f −2 ) 2

≥ ξ Mr,0 (FQ n ).

For the last inequality, the equality holds if and only if m = 0 or m = 22r −2kr + f −2 , that is ξ M r ,kr (FQ n ) = ξ M r ,0 (FQ n ) = ξ Mr,kr +1 (FQ n ). 4. The h-extra edge-connectivity Generally, the formulas (2) and (5) imply that the h-extra edge-connectivity of folded hypercube FQ n is

λh (FQ n ) =

min

h≤m≤2n−1

{(n + 1)m −

s 

t i 2t i −

i =0

where h = 1, 2, . . . , 2n−1 and m =

s 

s 

2 · i · 2t i },

(6)

i =0

2t i .

i =0

The following theorems of the h-extra edge-connectivity of FQ n can be easily deduced by the above formula and the properties of ξm (FQ n ). n

Theorem 4.1. For n ≥ 4 and h = 1, 2, . . . 2 2 +1 , the h-extra edge-connectivity of FQ n is

λh (FQ n ) =

⎧ s s   ⎪ n ⎪ ⎪ t i 2t i + 2i2t i ) for 1 ≤ h ≤ 2 2 +1 − dr ; ⎨(n + 1)h − ( ⎪ ⎪ n n ⎪ ⎩ 2 2 +1

i =0

i =0

otherwise,

2

where dr = 4 if n is odd and dr = 2 if n is even. n

Proof. Using the formula (6) and Lemma 3.4, for 1 ≤ h ≤ 2 2 +1 , n ≥ 4, we have n

λh (FQ n ) = min{ξm (FQ n ) : h ≤ m ≤ 2 2 +1 }. For 1 ≤ h =

s −1 i =0

n

2t i ≤ 2 2 +1 − dr , by Lemma 3.3, it is easy to see that n

λh (FQ n ) = min{ξm (FQ n ) : h ≤ m ≤ 2 2 +1 − dr }. By Corollary 3.2, s s   n λh (FQ n ) = min{ξm (FQ n ) : h ≤ m ≤ 2 2 +1 − dr } = ξh (FQ n ) = (n + 1)h − ( t i 2t i + 2i2t i ). i =0

 n2 +1

For 2

 n2 +1

− dr < h ≤ 2

, similarly, using the formula (6), Lemma 3.3 and Lemma 3.4,

n

n

λh (FQ n ) = min{ξm (FQ n ) : h ≤ m ≤ 2 2 +1 } = ξ So the proof is finished.

2

 n2 +1

n

(FQ n ) =  2 2 +1 . 2

2

The follow results can be obtained immediately from the Theorem 4.1. Corollary 4.2. (a) λ1 (FQ n ) = n + 1, 2 ≤ n [5]; (b) λ2 (FQ n ) = 2n, 2 ≤ n [23]; (c) λ3 (FQ n ) = 3n − 1, 4 ≤ n [24]; (d) λ4 (FQ n ) = 4n − 4, 5 ≤ n [4]; (e) λh (FQ n ) = (n + 1)h −

s 

i =0

t i 2t i +

s  i =0

2 · i · 2t i , for each h =

s  i =0

2t i , h ≤ n, 6 ≤ n [22].

i =0

M. Zhang et al. / Theoretical Computer Science 615 (2016) 71–77 n

n

Theorem 4.3. Suppose 2 2 +r − lr ≤ h ≤ 2 2 +r , where r = 1, 2, . . . ,  n2  − 1 and lr = Then

n

22r −1 3

77

if n is odd and lr =

22r +1 −2 3

if n is even.

n

λh (FQ n ) = (  − r + 1)2 2 +r . 2

n

n

Proof. For 2 2 +r − lr ≤ h ≤ 2 2 +r , by using the formula (6) and Lemma 3.5, we have

n

n

λh (FQ n ) = min{ξm (FQ n ) : h ≤ m ≤ 2 2 +r } = ξ

2

 n2 +r

n

(FQ n ) = (  − r + 1)2 2 +r . 2

2

n Corollary 4.4. When r =  n2  − 1, we obtain λh (FQ n ) = 2n , for  2 3+2  ≤ h ≤ 2n−1 .

5. Conclusions and further researches This paper deals with the h-extra edge-connectivity of folded hypercubes λh (FQ n ). We generalize some precious results (for h ≤ n) and give some new results. The exact values of λh (FQ n ) are given for 1 ≤ h ≤ 2n−1 except n n 1 for the cases of h’s satisfying that 2 2 +r −1 < h < 2 2 +r − lr , where r = 3, 4, . . . , n+ − 1 and lr = 2

odd and r = 2, 3, . . . ,

n 2

− 1, lr =

2

2r +1

3

−2

22r −1 3

if n is

if n is even. Although similarly by using the formula (6), Lemma 3.4 and n

Lemma 3.5, the formula λh (FQ n ) = min{ξm (FQ n ) : h ≤ m ≤ 2 2 +r − lr } works for the rest of h’s, the function ξm (FQ n ) n n for 2 2 +r −1 < m < 2 2 +r − lr has no monotonic property. For example, for n = 10, r = 3, ξ27 +25 −2 (FQ 10 ) = ξ27 +25 (FQ 10 ) < ξ27 +25 −1 (FQ 10 ), ξ27 +25 +24 −2 (FQ 10 ) = ξ27 +25 +24 (FQ 10 ) < ξ27 +25 +24 −1 (FQ 10 ), ξ27 +26 −2 (FQ 10 ) > ξ27 +26 −1 (FQ 10 ) > ξ27 +26 (FQ 10 ) but ξ27 +26 +24 −2 (FQ 10 ) = ξ27 +26 +24 (FQ 10 ) < ξ27 +26 +24 −1 (FQ 10 ). Hence, for the rest of h’s, before determining the λh (FQ n ), some new methods should be introduced to refine monotone interval of the function ξm (FQ n ). Besides, it would be interesting to improve the version of vertex cases, h-extra connectivity of FQ n , for g ≤ n + 2, n ≥ 7 by Mi Mi Zhang and Jin Xin Zhou in 2015 [27]. Acknowledgements We would like to thank the anonymous referees for their valuable suggestions and helpful comments. References [1] A.J. Boals, A.K. Gupta, N.A. Sherwani, Incomplete hypercubes: algorithms and embeddings, J. Supercomput. 8 (1994) 263–294. [2] C. Balbuena, P. García-Vázquez, Edge fault tolerance analysis of super k-restricted connected network, Appl. Math. Comput. 216 (2) (2010) 506–513. [3] H.L. Chen, N.F. Tzeng, A Boolean expression-based approach for maximum incomplete subcube identification in faulty hypercubes, IEEE Trans. Parallel Distrib. Syst. 8 (1997) 1171–1183. [4] N.-W. Chang, C.-Y. Tsai, S.-Y. Hsieh, On 3-extra connectivity and 3-extra edge connectivity of folded hypercubes, IEEE Trans. Comput. 63 (6) (2014) 1594–1600. [5] A. El-Amawy, S. Latifi, Properties and performance of folded hypercubes, IEEE Trans. Parallel Distrib. Syst. 2 (1) (1991) 31–42. [6] A.H. Esfahanian, Generalized measure of fault tolerance with application to n-cube networks, IEEE Trans. Comput. 38 (1989) 1586–1591. [7] J. Fàbrega, M.A. Fiol, On the extraconnectivity of graphs, Discrete Math. 155 (1996) 49–57. [8] F. Harary, Conditional connectivity, Networks 13 (1983) 346–357. [9] S.-Y. Hsieh, Y.-H. Chang, Extraconnectivity of k-ary n-cube networks, Theoret. Comput. Sci. 443 (2012) 63–69. [10] H. Katseff, Incomplete hypercubes, IEEE Trans. Comput. 37 (1988) 604–608. [11] S. Latifi, Simulation of PM21 network by folded hypercube, IEEE Proc. E Comput. Digit. Tech. 138 (6) (1991) 397–400. [12] S. Latifi, M. Hegde, M. Naraghi-Pour, Conditional connectivity measures for large multiprocessor systems, IEEE Trans. Comput. 43 (2002) 218–222. [13] H. Li, W. Yang, Bounding the size of the subgraph induced by m vertices and extra edge-connectivity of hypercubes, Discrete Appl. Math. 161 (16) (2013) 2753–2757. [14] J.S. Park, N.J. Davis IV, Modeling the folded hypercube ATM switches, in: The Proceedings of OPNETWORK 2001, Washington DC, 2001. [15] J.S. Park, N.J. Davis, The folded hypercube ATM switches, in: Proc. of IEEE International Conference on Networking, Part II, vol. 1, 2001, pp. 370–379. [16] J.X. Meng, Y. Ji, On a kind of restricted edge connectivity of graphs, Discrete Appl. Math. 243 (2002) 291–298. [17] I. Rajasingh, M. Arockiaraj, Linear wirelength of folded hypercubes, Math. Comput. Sci. 5 (2011) 101–111. [18] D. Wang, Embedding Hamiltonian cycles into folded hypercubes with faulty links, J. Parallel Distrib. Comput. 61 (2001) 545–564. [19] S. Wang, J. Yuan, A.X. Liu, K-restricted edge connectivity for some interconnection networks, Appl. Math. Comput. 201 (1–2) (2008) 587–596. [20] J.M. Xu, M. Ma, Cycles in folded hypercubes, Appl. Math. Lett. 19 (2006) 140–145. [21] J.M. Xu, Q. Zhu, X.M. Hou, T. Zhou, On restricted connectivity and extra connectivity of hypercubes and folded hypercubes, J. Shanghai Jiaotong Univ. 10 (2005) 203–207. [22] W. Yang, H. Li, On reliability of the folded hypercubes in terms of the extra edge-connectivity, Inform. Sci. 272 (2014) 238–243. [23] Q. Zhu, J.M. Xu, On restricted edge connectivity and extra edge connectivity of hypercubes and foled hypercubes, J. Univ. Sci. Technol. China 36 (2006) 246–253. [24] Q. Zhu, J.M. Xu, X. Hou, M. Xu, On reliability of the folded hypercubes, Inform. Sci. 177 (2007) 1782–1788. [25] Z. Zhang, Sufficient conditions for restricted-edge-connectivity to be optimal, Discrete Math. 307 (22) (2007) 2891–2899. [26] A. Hellwig, L. Volkmann, Sufficient conditions for graphs to be lambda’-optimal, super-edge-connected, and maximally edge-connected, J. Graph Theory 48 (3) (2005) 228–246. [27] M.M. Zhang, J.X. Zhou, On g-extra connectivity of folded hypercubes, Theoret. Comput. Sci. 593 (2015) 146–153.