The Maximum Wiener Index of Trees with Given Degree Sequences

0 downloads 0 Views 152KB Size Report
Jul 22, 2009 - The Wiener index of a molecular graph, introduced by Wiener [24] in 1947, is one of the oldest ... pending edges to a path v1,v2, ···,vk of length k 1 such that ... Shi in [22] proved that a maximum optimal tree must be a caterpillar. ..... Lemma 2.5 Let w1 ≥ w2 ≥···≥ wk ≥ 1 be the positive integers with k ≥ 5.

arXiv:0907.3772v1 [math.CO] 22 Jul 2009

The Maximum Wiener Index of Trees with Given Degree Sequences∗ Xiao-Dong Zhang†, Yong Liu and Min-Xian Han Department of Mathematics Shanghai Jiao Tong University 800 Dongchuan road, Shanghai, 200240, P.R. China

Abstract The Wiener index of a connected graph is the sum of topological distances between all pairs of vertices. Since Wang in [23] gave a mistake result on the maximum Wiener index for given tree degree sequence, in this paper, we investigate the maximum Wiener index of trees with given degree sequences and extremal trees which attain the maximum value.

Key words: Wiener index, tree, degree sequence, caterpillar. AMS Classifications: 05C12, 05C05, 05C90.

1

Introduction

The Wiener index of a molecular graph, introduced by Wiener [24] in 1947, is one of the oldest and most widely used topological indices in the quantitative structure property relationships. In the mathematical literature, the Wiener index seems to be the first studied by Entringer et al. [4]. For more information and background, ∗

Supported by National Natural Science Foundation of China (No.10531070), National Basic Research Program of China 973 Program (No.2006CB805900), National Research Program of China 863 Program (No.2006AA11Z209) and a grant of Science and Technology Commission of Shanghai Municipality (STCSM No: 09XD1402500). † Correspondent author: Xiao-Dong Zhang (Email: [email protected])

1

the readers may refer to a recent and very comprehensive survey [3] and a book [20] which is dedicated to Harry Wiener on the Wiener index and the references therein. Through this paper, all graphs are finite, simple and undirected. Let G = (V, E) be a simple connected graph with vertex set V (G) = {v1 , · · · , vn } and edge set E(G). Denote by dG (vi ) (or for short d(vi )) the degree of vertex vi . The distance between vertices vi and vj is the minimum number of edges between vi and vj and denoted by dG (vi , vj ) (or for short d(vi, vj )). The Wiener index of a connected graph G is defined as X W (G) = d(vi , vj ). (1) {vi ,vj }⊆V (G)

A tree is a connected and acyclic graph. A caterpillar is a tree in which a single path (called Spine) is incident to (or contains) every edge. For other terminology and notions, we follow from [1]. Entringer et al. [4] proved that the path Pn and the star K1,n−1 have the maximum and minimum Wiener indices, respectively, in the set consisting of all trees of order n. Dankelmann [2] obtained the all extremal graphs in the set of all connected graphs with given the order and the matching number which attained the maximum Wiener value. Moreover, Fischermann et al. [6] and Jelen et al. [14] independently determined all trees which have the minimum Wiener indices among all trees of order n and maximum degree ∆. A nonincreasing sequence of nonnegative integers π = (d1 , d2 , · · · , dn ) is called graphic if there exists a simple graph having π as its vertex degree sequence. Hence it is natural to consider the following problem. Problem 1.1 Let π = (d1 , · · · , dn ) be graphic degree sequence and Gπ = {G : the degree sequence of G is π}. Find the upper (lower) bounds for the Wiener index of all graphs G in Gπ and characterize all extremal graphs which attain the upper (lower) bounds. Moreover, we call a graph maximum (minimum) optimal if it maximizes (minimizes) the Wiener index in Gπ . Recently, by the different techniques, Wang [23] and Zhang et al.[25] independently characterized the tree that minimizes the Wiener index among trees of given degree sequences. Moreover, they proved that the minimum optimal trees for a given tree degree sequence π are unique. On the other hand, Wang in [23] also ”proved” the only maximum optimal tree that maximizes the Wiener index among trees of given degree sequences. The result can be stated as follows: 2

Theorem 1.2 [23] Given the degree sequence and the number of vertices, the greedy caterpillar maximizes the Wiener index, where the greedy caterpillar with degree sequence (d1 , · · · , dn ) (d1 ≥ d2 ≥ · · · ≥ dk ≥ 2 > dk+1 = 1) is formed by attaching pending edges to a path v1 , v2 , · · · , vk of length k − 1 such that d(v1 ) ≥ d(vk ) ≥ d(v2 ) ≥ d(vk−1 ) ≥ · · · ≥ d(v⌈ k+1 ⌉ ). 2

Unfortunately, this result is not correct. For example: Example 1.3 Let π = (13, 5, 5, 5, 4, 3, 1, · · · , 1) be a degree sequence of tree with 31 vertices. Let T1 and T2 be two trees with degree sequences π (see Fig.1). 12 2 3 3 4 c c c c c c c c c c c ··· ··· ··· ··· A A

A A

A c

 

A A

 

v1

A A

c

A c

v2

v3

 

A A

 

A A

A c

 

 

A A

v4

A A

A c



 

A A



A

v5

A  A c

 



v6

T1 c A A

12 ··· A A

A c

v1

 

 

c

c A A

3 ··· A A

A c

 

2 c  

c A A

v2

A A

A c

 

 

c

v3

c

c A A

3 ··· A A

c

A c

v4

v5



 



c

c A A

4 ··· A

A  A c

 

v6

T2 Figure 1 T1 and T2 Clearly, T2 is a greedy caterpillar and T1 is not a greedy caterpillar. Moreover, they have the same degree sequences π. By calculation, it is easy to see that W (T2 ) = 9870 < W (T1 ) = 9886. Hence this example illustrates that Theorem 1.2 in [23] is not correct. Motivated by Problem 1.1 and Example 1.3, we try to investigate the extremal trees which attain the maximum Wiener index among all trees with given degree sequences. The problem seems to be difficult. Because we find that the extremal tree depends on the values of components of degree sequences. The rest of the paper is organized as follows. In Section 2, we discuss some properties of the extremal tree with the maximum Wiener index and give an upper bound in terms of degree sequences. In Section 3, the extremal trees with the maximum Wiener index among given degree sequences (d1 , · · · , dn ), where d1 ≥ · · · ≥ dk ≥ 2 > dk+1 = 1 and k ≤ 6 are characterized. Moreover, the extremal maximal trees are not unique. 3



c

2

Properties of extremal trees with the maximum Wiener index

Let Tπ be the set of all trees with degree sequences π = (d1 , d2 , · · · , dn ) with d1 ≥ d2 ≥ · · · ≥ dn . Shi in [22] proved that a maximum optimal tree must be a caterpillar. Lemma 2.1 [22] Let T ∗ be a maximum optimal tree in Tπ . Then T ∗ is a caterpillar. From Lemma 2.1, we only need to consider all caterpillars with a degree sequence π. In order to study the structure of the maximum optimal trees, we present a formula for Wiener index of any caterpillar. Lemma 2.2 Let T be a caterpillar of order n with the degree sequence π = (d(v1 ), · · · , d(vk ), d(vk+1 ), · · · , d(vn ))(see Figure 2).

c A A

y1 ··· A A

A c

v1

 

 

c

y2 − 1 ···

c A A

A A

A c

v2

 

 

c

c A A

yi − 1 c ··· A A

 

−1

y

 

· · · A c · · · vi Figure 2 T

c k−1 c ··· A   A  A A  A c

c A A

yk ··· A A

vk−1

 A c

 



c

vk

If d(vi ) = yi + 1 ≥ 2 for i = 1, · · · , k and d(vk+1 ) = · · · = d(vn ) = 1, then W (T ) = (n − 1)2 + F (y1 , · · · , yk ), where F (y1 , · · · , yk ) =

k−1 X

(

i X

i=1 j=1

yj )(

k X

yj ).

(2)

(3)

j=i+1

Proof. It is well known [12] that the formula (1) is equal to W (T ) =

X

n1 (e)n2 (e),

e

where e = (u, v) is an edge of T , and n1 (e) (resp. n2 (e)) is the number of vertices of the component of T − e containing u (resp. v). For ei = (vi , vi+1 ) ∈ E(T ), the

4

numbers of vertices of the two components of T − ei are ij=1 d(vj ) − (i − 1) and Pk j=i+1 d(vj ) − (k − i − 1) for i = 1, · · · , k − 1, respectively. Hence P

W (T ) =

X

n1 (e)n2 (e)

e∈E(T )

=

X

X

n1 (e)n2 (e) +

e is pendent edge

n1 (e)n2 (e)

e is not pendent edge

= (n − 1)(n − k) +

k−1 X

(

i X

k−1 X

(1 +

i X

yj )(1 +

= (n − 1)(n − k) + (k − 1)(1 +

k X

yj )

j=i+1

j=1

i=1

d(vj ) − (k − i − 1))

j=i+1

i=1 j=1

= (n − 1)(n − k) +

k X

d(vj ) − (i − 1))(

k X

yj ) +

k−1 X

(

i X

yj )(

yj )

j=i+1

i=1 j=1

j=1

k X

= (n − 1)2 + F (y1, · · · , yk ), where last equality is due to This completes the proof.

Pk

j=1

yj =

Pk

j=1 d(vj ) −k

= 2(n−1) −(n−k) −k = n−2.

Remark In this sequel, the caterpillar T in Lemma 2.2 is denoted by T (y1 , · · · , yk ). Then degree sequence of T (y1, · · · , yk ) is (y1 + 1, · · · , yk + 1, 1, · · · , 1). The following theorem give a characterization of a maximum optimal tree. Theorem 2.3 Let π = (d1 , · · · , dn ) with d1 ≥ · · · ≥ dk ≥ 2 ≥ dk+1 = · · · = dn = 1. Then T is a maximum optimal tree in Tπ if and only if T is a caterpillar T (x1 , · · · , xk ) and (x1 , · · · , xk ) satisfies F (x1 , · · · , xk ) = max{F (y1, · · · , yk ) =

k−1 X

(

i X

i=1 j=1

yj )(

k X

yj ) : y1 ≥ yk },

(4)

j=i+1

where (y1 , · · · , yk ) is any permutation of (d1 − 1, · · · , dk − 1). Proof. Necessity. Since T is a maximum optimal tree in Tπ , by Lemmas 2.1, T must be a caterpillar and can be denoted by T (z1 , · · · , zk ) with (z1 , · · · , zk ) is the permutation of (d1 − 1, · · · , dk − 1). Moreover, by Lemma 2.2, we have W (T (z1 , · · · , zk )) = (n − 1)2 + F (z1 , · · · , zk ). For any permutation (y1 , · · · , yk ) of (d1 − 1, · · · , dk − 1) with y1 ≥ yk , there exists a caterpillar T1 with the degree sequence π such that W (T1 ) = (n − 1)2 + F (y1 , · · · , yk ). 5

Because T (z1 , · · · , zk ) is a maximum optimal tree in Tπ , we have F (y1, · · · , yk ) = W (T1 ) − (n − 1)2 ≤ W (T (z1 , · · · , zk )) − (n − 1)2 = F (z1 , · · · , zk ). Sufficiency. If T is a caterpillar T (x1 , · · · , xk ) and (x1 , · · · , xk ) satisfies F (x1 , · · · , xk ) = max{F (y1 , · · · , yk ) =

k−1 X

(

i X

yj )(

k X

yj ) : y1 ≥ yk },

(5)

j=i+1

i=1 j=1

where the maximum is taken over all permutations (y1, · · · , yk ) of (d1 − 1, · · · , dk − 1). Let T1 be any tree with the degree sequence π. By Lemma 2.1, there exists a caterpillar T2 with the degree sequence π such that W (T1 ) ≤ W (T2 ). Then T2 must be T (y1 , · · · , yk ), where (y1 , · · · , yk ) is the permutation of (d1 − 1, · · · , dk − 1). Hence W (T1 ) ≤ W (T2 ) = (n−1)2 +F (y1 , · · · , yk ) ≤ (n−1)2 +F (x1 , · · · , xk ) = W (T (x1 , · · · , xk )). Therefore T (x1 , · · · , xk ) is a maximum optimal tree. This completes the proof. Now we can present an upper bound for the Wiener index of any tree with given degree sequence π in terms of degree sequences. Theorem 2.4 Let T be a tree with a given degree sequence π = (d1 , · · · , dn ), where d1 ≥ · · · ≥ dk > dk+1 = · · · = dn = 1. Then W (T ) ≤ (n − 1)2 +

k k(k − 1) X (di − 1)2 4 i=1

(6)

with equality if and only if k = 2 and d1 = d2 . Proof. Let T (x1 , · · · , xk ) be a caterpillar and (x1 , · · · , xk ) satisfy F (x1 , · · · , xk ) = max{F (y1 , · · · , yk ) =

k−1 X

(

i X

i=1 j=1

yj )(

k X

yj ) : y1 ≥ yk },

(7)

j=i+1

where (y1 , · · · , yk ) is any permutation of (d1 −1, · · · , dk −1). By Theorem 2.3, W (T ) ≤ W (T (x1 , · · · , xk )). Clearly, F (x1 , · · · , xk ) =

k−1 X

(

i X

i=1 j=1

where



C=

     

xj )(

k X

1 xj ) = (x1 , · · · , xk )C(x1 , · · · , xk )T , 2 j=i+1

0 1 2 1 0 1 ··· ··· ··· k−1 k−2 k−3 6

··· k −2 k −1 ··· k −3 k −2 ··· ··· ··· ··· 1 0



   .  

By Perron-Frobenius theorem (for example, see [11]), the largest eigenvalue λ1 (C) of C is at most k(k−1) with equality if and only if k = 2. Hence by Rayleigh quotient, 2 (x1 , · · · , xk )C(x1 , · · · , xk )T ≤ λ1 (C)

k X

x2i

i=1

with equality if and only if (x1 , · · · , xk )T is an eigenvector of C corresponding to the eigenvalue λ1 (C). Therefore, F (x1 , · · · , xk ) ≤

k k(k − 1) X x2i 4 i=1

with equality if and only if k = 2 and x1 = x2 . Hence W (T ) ≤ (n − 1)2 +

k k k(k − 1) X k(k − 1) X xi 2 ≤ (n − 1)2 + (di − 1)2 4 4 i=1 i=1

with equality if and only if k = 2 and d1 = d2 , since (d(v1 ), · · · , d(vk )) is a permutation of (d1 , · · · , dk ). This completes the proof. Lemma 2.5 Let w1 ≥ w2 ≥ · · · ≥ wk ≥ 1 be the positive integers with k ≥ 5. Let F (z1 , · · · , zk ) = max{F (y1, · · · , yk ) =

k−1 X

(

i X

i=1 j=1

yj )(

k X

yj ) : y1 ≥ yk },

j=i+1

where (y1 , · · · , yk ) is any permutation of (w1 , · · · , wk ). Then there exists a 2 ≤ t ≤ k − 2 such that the following holds: z1 + · · · + zt−2 ≤ zt+1 + · · · + zk

(8)

z1 + · · · + zt−1 > zt+2 + · · · + zk .

(9)

and Further, if equations (8) is strict, then z1 ≥ z2 ≥ · · · ≥ zt ,

zt ≤ zt+1 ≤ · · · ≤ zk .

(10)

zt ≤ zt+1 ≤ · · · ≤ zk

(11)

If equations (8) becomes equality, then z1 ≥ z2 ≥ · · · ≥ zt , or z1 ≥ z2 ≥ · · · ≥ zt−1 ,

zt−1 ≤ zt ≤ · · · ≤ zk . 7

(12)

Proof. Let f (p) =

p−2 X

zi −

k X

zi ,

2 ≤ p ≤ k − 2.

i=p+1

i=1

Clearly f (2) < 0, f (k − 1) > 0 and f (2) ≤ f (3) ≤ · · · ≤ f (k − 1). Hence there exists a 2 ≤ t ≤ k − 2 such that f (t) ≤ 0 and f (t + 1) > 0. In other words, equations (8) and (9) hold. By the definition of F (z1 , · · · , zk ), we have for 1 ≤ i ≤ k − 1, 0 ≤ F (z1 , · · · , zi−1 , zi , zi+1 , · · · , zk ) − F (z1 , · · · , zi−1 , zi+1 , zi , · · · , zk ) = (zi+1 − zi )(

i−1 X

j=1

zj −

k X

zj ).

j=i+2

k But for 1 ≤ i ≤ t − 2, by (8), we have i−1 j=1 zj < j=i+2 zj . Hence z1 ≥ · · · ≥ zt−1 . P Pk On the other hand, for t ≤ i ≤ k − 1, by (9), we have i−1 j=1 zj > j=i+2 zj . Therefore zt ≤ zt+1 · · · ≤ zk . If (8) is strict, then (z1 + · · ·+ zt−2 ) − (zt+1 + · · · + zk ) < 0, which implies zt−1 ≥ zt . So (10) holds. If (8) becomes equality, i.e., z1 + · · · + zt−2 = zt+1 + · · · + zk , then it is easy to see that (11) or (12) holds. This completes the proof.

P

P

Corollary 2.6 Let w1 ≥ w2 ≥ · · · ≥ w6 ≥ 1 be the positive integers. Let F (z1 , · · · , z6 ) = max{F (y1 , · · · , y6 ) =

i 5 X X

(

i=1 j=1

yj )(

6 X

yj ) : y1 ≥ y6 },

j=i+1

where (y1 , · · · , y6 ) is any permutation of (w1 , · · · , w6). Then (z1 , · · · , z6 ) is equal to one of the following five (w1 , w6, w5 , w4 , w3 , w2 ), (w1 , w5 , w6, w4 , w3 , w2 ), (w1 , w4 , w6 , w5, w3 , w2 ), (w1 , w4 , w5 , w6 , w3 , w2) and (w1 , w3 , w6, w5 , w4 , w2 ). Proof. By Lemma 2.5, there are just three cases: Case 1 t = 2. Then by Lemma 2.5, z1 ≥ z2 and z2 ≤ z3 ≤ z4 ≤ z5 ≤ z6 . Hence (z1 , · · · , z6 ) must be (w1 , w6 , w5, w4 , w3 , w2 ). Case 2 t = 3. Then z1 ≤ z4 + z5 + z6 and z1 + z2 > z5 + z6 . Moreover, z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 ; or z1 ≥ z2 and z2 ≤ z3 ≤ z4 ≤ z5 ≤ z6 . Therefore (z1 , · · · , z6 )

8

must be one of (w1 , w6 , w5 , w4, w3 , w2 ), (w1 , w5 , w6 , w4 , w3 , w2 ), (w1 , w4 , w6, w5 , w3 , w2 ) and (w1 , w3 , w6 , w5, w4 , w2 ). Case 3 t = 4. Then z1 + z2 ≤ z5 + z6 . Moreover, z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 ; or z1 ≥ z2 ≥ z3 ≥ z4 and z4 ≤ z5 ≤ z6 . Therefore, (z1 , · · · , z6 ) must be one of (w1 , w4, w6 , w5 , w3 , w2 ), (w1 , w5, w6 , w4 , w3 , w2 ) and (w1 , w4, w5 , w6 , w3 , w2 ). This completes the proof. Theorem 2.7 Let π = (d1 , · · · , dn ) be a tree degree sequence with d1 ≥ d2 ≥ · · · ≥ dk ≥ 2, dk+1 = · · · = dn = 1 and k ≥ 5. If a caterpillar T (x1 , · · · , xk ) is a maximum optimal tree in Tπ with F (x1 , · · · , xk ) in equation (2). Then there exists a 2 ≤ t ≤ k−2 such that either t−2 X

xi ≤

k X

xi ,

xi >

k X

xi ,

x1 ≥ x2 ≥ · · · ≥ xt−1 ≥ xt ,

xt ≤ xt+1 ≤ · · · ≤ xk ;

k X

xi ,

x1 ≥ x2 ≥ · · · ≥ xt−1 ≥ xt ,

xt ≤ xt+1 ≤ · · · ≤ xk ;

t+2

i=1

i=t+1

i=1

t−1 X

or t−2 X

xi =

i=1

k X

xi ,

i=t+1

t−1 X

xi >

i=1

t+2

or t−2 X i=1

xi =

k X

i=t+1

xi ,

t−1 X i=1

xi >

k X

xi ,

x1 ≥ x2 ≥ · · · ≥ xt−1 ,

xt−1 ≤ xt ≤ · · · ≤ xk .

t+2

Proof. It follows from Theorem 2.3 and Lemma 2.5 that the assertion holds.

3

The maximum optimal tree with many leaves

In this section, for a given degree sequence π = (d1 , · · · , dn ) with at least n − 6 leaves, we give the maximum optimal trees with the maximum Wiener index in Tπ . Moreover, the maximum optimal tree may be not unique. Theorem 3.1 Let π = (d1 , · · · , dk , · · · , dn ) be tree degree sequence with n − k leaves for 2 ≤ k ≤ 4. Then the maximum optimal tree in Tπ is the greedy caterpillar. In other words, if k = 2, then W (T ) = (n − 1)2 + (d1 − 1)(d2 − 1), for T ∈ Tπ . If k = 3, then for any T ∈ Tπ , W (T ) ≤ (n − 1)2 + (d1 − 1)(d2 + d3 − 2) + (d1 + d2 − 2)(d3 − 1) 9

with equality if and only if T is the caterpillar T (d1 − 1, d3 − 1, d2 − 1). If k = 4, then for any T ∈ Tπ , W (T ) ≤ (n−1)2 +(d1 −1)(d2 +d3 +d4 −3)+(d1 +d2 −2)(d3 +d4 −2)+(d1 +d2 +d3 −3)(d4 −1) with equality if and only if T is the caterpillar T (d1 − 1, d4 − 1, d3 − 1, d2 − 1). Proof. If k = 2, it is obvious. If k = 3, it is easy to see that F (d1 −1, d2 −1, d3 −1) ≤ F (d1 − 1, d3 − 1, d2 − 1). By Theorem 2.3, the assertion holds. If k = 4, then by Theorem 2.3, let T be a caterpillar T (x1 , x2 , x3 , x4 ) and F (x1 , x2 , x3 , x4 ) = max{F (y1 , y2, y3 , y4 ) : y1 ≥ y4 }, where (y1 , y2, y3 , y4 ) is any permutation of (d1 − 1, d2 − 1, d3 − 1, d4 − 1). Because F (x1 , x2 , x3 , x4 ) − F (x2 , x1 , x3 , x4 ) = (x1 − x2 )(x3 + x4 ) ≥ 0 and F (x1 , x2 , x3 , x4 ) − F (x1 , x2 , x4 , x3 ) = (x4 − x3 )(x1 + x2 ) ≥ 0, we have x1 ≥ x2 and x4 ≥ x3 . So (x1 , x2 , x3 , x4 ) = (d1 − 1, d4 − 1, d3 − 1, d2 − 1). This completes the proof. Theorem 3.2 Let π = (d1 , · · · , dk , · · · , dn ) be tree degree sequence with n − 5 leaves. (1). If d1 > d2 + d3 , then the maximum optimal tree in Tπ is the only caterpillar T (d1 − 1, d5 − 1, d4 − 1, d3 − 1, d2 − 1). (2). If d1 = d2 + d3 , then there are the exactly two maximum optimal trees in Tπ : one tree is the caterpillar T (d1 − 1, d5 − 1, d4 − 1, d3 − 1, d2 − 1); the other tree is the caterpillar T (d1 − 1, d4 − 1, d5 − 1, d3 − 1, d2 − 1). (3). If d1 < d2 + d3 , then the maximum optimal tree in Tπ is the only caterpillar T (d1 − 1, d4 − 1, d5 − 1, d3 − 1, d2 − 1). Proof. By Theorem2.3, let T (x1 , x2 , x3 , x4 , x5 ) be a maximum optimal tree in Tπ . If d1 > d2 + d3 , then by Theorem 2.7, it is easy to see that t = 2, and x1 ≥ x2 and x2 ≤ x3 ≤ x4 ≤ x5 . Hence (x1 , x2 , x3 , x4 , x5 ) = (d1 − 1, d5 − 1, d4 − 1, d3 − 1, d2 − 1). If d1 < d2 + d3 , then by Theorem 2.7, it is easy to see that x1 ≥ x2 ≥ x3 and x3 ≤ x4 ≤ x5 . Hence (x1 , x2 , x3 , x4 , x5 ) = (d1 − 1, d4 − 1, d5 − 1, d3 − 1, d2 − 1) or (d1 − 1, d3 − 1, d5 − 1, d4 − 1, d2 − 1). But W (T (d1 − 1, d4 − 1, d5 − 1, d3 − 1, d2 − 1)) − 10

W (T (d1 − 1, d3 − 1, d5 − 1, d4 − 1, d2 − 1)) = 2(d1 − d2 )(d3 − d4 ) ≥ 0 with equality if and only if d1 = d2 or d3 = d4 . Hence the assertion (3) holds. If d1 = d2 + d3 , then by Theorem 2.7, it is easy to see that either x1 ≥ x2 and x2 ≤ x3 ≤ x4 ≤ x5 ; or x1 ≥ x2 ≥ x3 and x3 ≤ x4 ≤ x5 . Hence (x1 , x2 , x3 , x4 , x5 ) = (d1 − 1, d5 − 1, d4 − 1, d3 − 1, d2 − 1) or (d1 − 1, d4 − 1, d5 − 1, d3 − 1, d2 − 1). Moreover, F (d1 − 1, d5 − 1, d4 − 1, d3 − 1, d2 − 1) = F (d1 − 1, d4 − 1, d5 − 1, d3 − 1, d2 − 1). Hence (2) holds. Lemma 3.3 Let w1 ≥ w2 ≥ · · · ≥ w6 ≥ 1 be positive integers and F (y1 , · · · , yk ) =

k−1 X

(

i X

i=1 j=1

yj )(

k X

yj ).

j=i+1

Then F (w1 , w6 , w5 , w4, w3 , w2 ) − F (w1, w5 , w6 , w4 , w3 , w2 ) = (w1 − w2 − w3 − w4 )(w5 − w6 ), (13) F (w1 , w5, w6 , w4 , w3 , w2 )−F (w1, w4 , w6 , w5 , w3 , w2 ) = 2(w1 −w2 −w3 )(w4 −w5 ), (14) F (w1 , w4 , w6 , w5, w3 , w2 ) − F (w1 , w4 , w5, w6 , w3 , w2 ) = (w1 + w4 − w2 − w3 )(w5 − w6 ), (15) F (w1 , w4 , w5 , w6, w3 , w2 ) −F (w1 , w3 , w6 , w5 , w4 , w2 ) = (3w3 −3w4 −w5 + w6 )(w1 −w2 ). (16) Proof. By a simple calculation, it is easy to see that the assertion holds. Theorem 3.4 Let π = (d1 , · · · , d6 , · · · , dn ) be tree degree sequence with n − 6 leaves, i.e., d1 ≥ · · · ≥ d6 ≥ 2 and d7 = · · · = dn = 1. (1). If d1 > d2 + d3 + d4 − 2, then there is only one maximum optimal tree T (d1 − 1, d6 − 1, d5 − 1, d4 − 1, d3 − 1, d2 − 1) in Tπ . (2). If d1 = d2 + d3 + d4 − 2, then there are exactly two maximum optimal trees in Tπ : one maximum optimal tree is T (d1 − 1, d6 − 1, d5 − 1, d4 − 1, d3 − 1, d2 − 1); the other maximum optimal tree is T (d1 − 1, d5 − 1, d6 − 1, d4 − 1, d3 − 1, d2 − 1). (3). d2 + d3 − 1 < d1 < d2 + d3 + d4 − 2, then there is only one maximum optimal tree T (d1 − 1, d5 − 1, d6 − 1, d4 − 1, d3 − 1, d2 − 1) in Tπ . (4). If d2 + d3 − 1 = d1 , then there are exactly two maximum optimal trees in Tπ : one maximum optimal tree is T (d1 − 1, d5 − 1, d6 − 1, d4 − 1, d3 − 1, d2 − 1); the other maximum optimal tree is T (d1 − 1, d4 − 1, d6 − 1, d5 − 1, d3 − 1, d2 − 1). 11

(5). If max{d2 + d3 − d4 , d2 + 13 (d5 − d6 )} < d1 < d2 + d3 − 1, then there is only one maximum optimal tree T (d1 − 1, d4 − 1, d6 − 1, d5 − 1, d3 − 1, d2 − 1) in Tπ . (6). If d1 = d2 + d3 − w4 > d2 + 31 (d5 − d6 ), then there are exactly two maximum optimal trees in Tπ : one maximum optimal tree is T (d1 − 1, d4 − 1, d6 − 1, d5 − 1, d3 − 1, d2 −1); the other maximum optimal tree is T (d1 −1, d4 −1, d5 −1, d6 −1, d3 −1, d2 −1). (7). If d1 = d2 + 31 (d5 − d6 ) > d2 + d3 − d4 , then there are exactly two maximum optimal trees in Tπ : one maximum optimal tree is T (d1 − 1, d4 − 1, d6 − 1, d5 − 1, d3 − 1, d2 −1); the other maximum optimal tree is T (d1 −1, d3 −1, d6 −1, d5 −1, d4 −1, d2 −1). (8). If d1 = d2 + d3 − d4 = d2 + 13 (d5 − d6 ), then there are exactly three maximum optimal trees in Tπ : they are T (d1 − 1, d4 − 1, d6 − 1, d5 − 1, d3 − 1, d2 − 1); T (d1 − 1, d4 − 1, d5 − 1, d6 − 1, d3 − 1, d2 − 1) and T (d1 − 1, d3 − 1, d6 − 1, d5 − 1, d4 − 1, d2 − 1). (9). If d2 + 13 (d5 −d6 ) ≤ d1 < d2 +d3 −d4 , or d1 ≤ d2 + 13 (d5 −d6 ) < d2 +d3 −d4 , then there is only one maximum optimal tree T (d1 − 1, d4 − 1, d5 − 1, d6 − 1, d3 − 1, d2 − 1) in Tπ . (10). If d2 + d3 − d4 ≤ d1 < d2 + 31 (d5 − d6 ); or d1 ≤ d2 + d3 − d4 < d2 + 31 (d5 − d6 ), then there is only one maximum optimal tree T (d1 −1, d3 −1, d6 −1, d5 −1, d4 −1, d2 −1) in Tπ . (11). If d1 < d2 + 13 (d5 − d6 ) = d2 + d3 − d4 , then there are exactly two maximum optimal trees in Tπ : one maximum optimal tree is T (d1 − 1, d3 − 1, d6 − 1, d5 − 1, d4 − 1, d2 −1); the other maximum optimal tree is T (d1 −1, d4 −1, d5 −1, d6 −1, d3 −1, d2 −1). Proof. The proof is referred to appendix since it is technique. Remark. From Theorem 3.4, we can see that the maximum optimal trees depend on the values of all components of the tree degree sequences and not unique, while the minimum optimal tree is unique for a given tree degree sequence. Moreover, Theorem 3.4 explains that it seems to be difficult for characterize all the maximum optimal trees for a given tree degree sequence.

References [1] J. A. Bondy and U. S. R. Murty, Graph theory with applications, Macmillan Press, New York, 1976. [2] P. Dankelmann, Average distance and independence number, Discrete Applied Mathematics, 51(1994) 75-83. 12

[3] A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66(2001) 211-249. [4] R. C. Entringer, D. E. Jackson and D. A. Synder, Distance in graphs, Czechoslovak Math. J. 26(1976) 283-296. [5] P. Erd¨os and T. Gallai, Graphs with prescribed degrees of vertices (Hungarian) Mat. Lapok, 11(1960) 264-274. [6] M. Fischermann, A. Hoffmann, D. Rautenbach, L. Szekely and L. Volkmann, Wiener index versus maximum degree in trees, Discrete Applied Mathematics, 122(2002) 127-137. [7] M. Fischermann, D. Rautenbach and L. Volkmann, Extremal trees with respect to dominance order, Ars Comb. 76(2005) 249-255. ˇ Tomovi´c, The multilicative version of [8] I. Gutman, W. Linert, I. Lukovits and Z the Wiener index, J. Chem. Inf. Comput. Sci. 40(2000) 113-116. [9] I. Gutman and J. H. Potgieter, Wiener index and intermolecular forces, J. Serb. Chem. Soc. 62(1997) 185-192. [10] I. Gutman, Y. N. Yeh, S. L. Lee and J. C. Chen, Wiener numbers of dendrimers, Comm. Math. Chem. (MATCH) 30(1994) 103-115. [11] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, London, 1985. [12] H. Hosoya, Topological index, A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn. 4(1971) 2332-2339. [13] D. J. Klein and D. Babiacutec, Partial orderings in chemistry, J. Chem. Inf. and Comp. Sci. 37(1997) 656-671. [14] F. Jelen and E. Triesch, Superdominance order and distance of trees with bounded maximum degree, Discrete Applied Mathematics, 125(2003) 225-233. [15] L. Lov´asz, Combinatorial Problems and Exercises, 2nd Edition, North-Holland, Amsterdam, 1993. 13

[16] I. Lukovits, General formulas for the Wiener index, J. Chem. Inf. Comput. Sci. 31(1991) 503-507. [17] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra and Applications, 197-198 (1994): 143-176. [18] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and engineering, vol.143, Academic Proess, New York, 1979. [19] O. E. Polansky and D. Bonchev, The Wiener number of graphs. I. general theory and changes due to some graph operations MATCH Commun. Math. Comput. Chem. 21(1986) 133-186. [20] D. H. Rouvray and R. B.King, Topology in Chemistry, Horwood Pub., Chichester, 2002. [21] E. Ruch and I. Gutman, The branching extent of graphs, J. Comb. Inf. and System Sci. 4(1979) 285-295. [22] R. Shi, The average distance of trees, Systems Science and Mathematical Sciences, 6(1)(1993), 18-24. [23] H. Wang, The extremal values of the Wiener index of a tree with given degree sequence, Discrete Applied Mathematics, 156(2009) 2647-2654. [24] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69(1947), 17-20. [25] X. D. Zhang, The Wiener index of trees with given degree sequences, MATCH Commun. Math. Comput. Chem., 60 (2008) 623-644.

14

Appendix: Proof of Theorem 3.4 Lemma 3.5 Let w1 ≥ w2 ≥ · · · ≥ w6 ≥ 1 be positive integers. If F (z1 , · · · , z6 ) = max{F (y1 , · · · , y6 ) : y1 ≥ y6 }, where (y1 , · · · , y6 ) is any permutation of (w1 , · · · , w6 ), then the following statement holds. (1). If w1 > w2 + w3 + w4 , then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w6 , w5, w4 , w3 , w2 ). (2). If w1 = w2 + w3 + w4 , then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w6 , w5 , w4 , w3 , w2) or (w1 , w5 , w6 , w4 , w3 , w2). (3). w2 +w3 < w1 < w2 +w3 +w4 , then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w5 , w6 , w4 , w3 , w2 ). (4). If w2 +w3 = w1 , then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w5 , w6 , w4 , w3 , w2 ) or (w1 , w4 , w6 , w5 , w3 , w2 ). (5). If max{w2 +w3 −w4 , w2 + 31 (w5 −w6 )} < w1 < w2 +w3 , then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w4 , w6 , w5 , w3 , w2). (6). If w1 = w2 + w3 − w4 > w2 + 31 (w5 − w6 ), then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w4 , w6 , w5 , w3 , w2) or (w1 , w4 , w5 , w6 , w3 , w2). (7). If w1 = w2 + 31 (w5 − w6 ) > w2 + w3 − w4 , then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w4 , w6 , w5 , w3 , w2) or (w1 , w3 , w6 , w5 , w4 , w2). (8). If w1 = w2 + w3 − w4 = w2 + 31 (w5 − w6 ), then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w4 , w6 , w5 , w3 , w2), or (w1 , w4 , w5 , w6 , w3 , w2) or (w1 , w3 , w6 , w5 , w4 , w2). (9). If w2 + 31 (w5 −w6 ) ≤ w1 < w2 +w3 −w4 , or w1 ≤ w2 + 31 (w5 −w6 ) < w2 +w3 −w4 , then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w4 , w5 , w6 , w3, w2 ). (10). If w2 + w3 − w4 ≤ w1 < w2 + 31 (w5 − w6 ); or w1 ≤ w2 + w3 − w4 < w2 + 31 (w5 − w6 ), then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w3 , w6 , w5 , w4 , w2 ). (11). If w1 < w2 + 31 (w5 − w6 ) = w2 + w3 − w4 , then (z1 , z2 , z3 , z4 , z5 , z6 ) = (w1 , w4 , w5 , w6 , w3 , w2) or (w1 , w3 , w6 , w5 , w4 , w2). Proof. (1). w1 > w2 + w3 + w4 . By (8) and (9) in Lemma 2.5, we have t = 2 and (z1 , · · · , z6 ) = (w1 , w6 , w5 , w4 , w3, w2 ). (2). w1 = w2 + w3 + w4 . By (8) and (9) in Lemma 2.5, we have t = 3. By (11) and (12). we consider the following two cases. If z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 , then by corollary 3.5 and w1 = w2 +w3 +w4 , we have (z1 , · · · , z6 ) = (w1 , w5, w6 , w4 , w3 , w2 ). If z1 ≥ z2 and z2 ≤ z3 ≤ z4 ≤ z5 ≤ z6 , then (z1 , · · · , z6 ) = (w1 , w6, w5 , w4 , w3 , w2 ). Hence (2) holds. (3). w2 + w3 < w1 < w2 + w3 + w4 . We consider the following four cases: 15

Case 1: w2 + w3 + w5 < w1 < w2 + w3 + w4 . By (8) and (9) in Lemma 2.5, we have t = 3 and z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 . Hence by Corollary 3.5, (z1 , · · · , z6 ) = (w1 , w5 , w6 , w4 , w3, w2 ). Case 2: w2 +w3 +w5 = w1 < w2 +w3 +w4 . Similarly, (z1 , · · · , z6 ) = (w1 , w5 , w6 , w4 , w3 , w2 ). Case 3: w2 + w4 + w5 < w1 < w2 + w3 + w5 and w1 > w2 + w3 . By (8) and (9) in Lemma 2.5, we have t = 3. Further (z1 , · · · , z6 ) = (w1 , w5 , w6 , w4 , w3, w2 ) or (z1 , · · · , z6 ) = (w1 , w4 , w6 , w5 , w3, w2 ). But by Lemma 3.3, we have F (w1 , w5 , w6 , w4, w3 , w2 ) − F (w1 , w4 , w6, w5 , w3 , w2 ) = 2(w1 − w2 − w3 )(w4 − w5 ). Hence (z1 , · · · , z6 ) = (w1 , w5 , w6 , w4, w3 , w2 ). Case 4: w2 + w3 < w1 ≤ w2 + w4 + w5 . By (8) and (9) in Lemma 2.5, we have t = 3. Further (z1 , · · · , z6 ) = (w1 , w3 , w6 , w5, w4 , w2 ), or (w1 , w4, w6 , w5 , w3 , w2 ), or (w1 , w5 , w6 , w4 , w3 , w2). But by Lemma 3.3, we have F (w1 , w3 , w6 , w5, w4 , w2 ) − F (w1 , w2 , w6 , w5 , w4 , w3 ) = 2(w2 − w3 )(2w1 − w4 + w6 ) ≥ 0, F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (w3 −w4 )(3w1−3w2 −w5 +w6 ) ≥ 0 and F (w1 , w5 , w6 , w4, w3 , w2 ) − F (w1 , w4 , w6, w5 , w3 , w2 ) = 2(w1 − w2 − w3 )(w4 − w5 ). Hence (z1 , · · · , z6 ) = (w1 , w5 , w6 , w4, w3 , w2 ). (4). w2 + w3 = w1 . From the proof of (3), it is easy to see that (z1 , · · · , z6 ) = (w1 , w5 , w6 , w4 , w3 , w2) or (w1 , w4, w6 , w5 , w3 , w2 ), because F (w1 , w5, w6 , w4 , w3 , w2 ) − F (w1 , w4 , w6 , w5, w3 , w2 ) = 0. Therefore (4) holds. (5). max{w2 + w3 − w4 , w2 + 31 (w5 − w6 )} < w1 < w2 + w3 . We consider the four cases. Case 1: w1 > w2 +w4 +w5 and w1 > w2 +w3 −w5 . By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 . Then (z1 , · · · , z6 ) = (w1 , w5 , w6, w4 , w3 , w2 ) or (w1 , w4 , w6 , w5 , w3 , w2 ). But F (w1 , w5, w6 , w4 , w3 , w2 ) − F (w1, w4 , w6 , w5 , w3 , w2 ) = 2(w4 − w5 )(w1 − w2 − w3 ) ≤ 0 with equality if and only if w4 = w5 . Therefore (z1 , · · · , z6 ) = (w1 , w4, w6 , w5 , w3 , w2 ). Case 2: w1 > w2 +w4 +w5 and w1 ≤ w2 +w3 −w5 . By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 . Then (z1 , · · · , z6 ) = (w1 , w5 , w6, w4 , w3 , w2 ) or (w1 , w4 , w6 , w5 , w3 , w2 ). But by Lemma 3.3, we have F (w1 , w5, w6 , w4 , w3 , w2 ) − F (w1, w4 , w6 , w5 , w3 , w2 ) = 2(w4 − w5 )(w1 − w2 − w3 ) ≤ 0 16

with equality if and only if w4 = w5 . Therefore (z1 , · · · , z6 ) = (w1 , w4, w6 , w5 , w3 , w2 ). Case 3: w1 ≤ w2 +w4 +w5 and w1 > w2 +w3 −w5 . By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 . Then (z1 , · · · , z6 ) = (w1 , w5, w6 , w4 , w3 , w2 ), or (w1 , w4 , w6 , w5 , w3 , w2 ), or (w1 , w3, w6 , w5 , w4 , w2 ). But by Lemma 3.3, we have F (w1 , w5, w6 , w4 , w3 , w2 ) − F (w1, w4 , w6 , w5 , w3 , w2 ) = 2(w4 − w5 )(w1 − w2 − w3 ) ≤ 0 with equality if and only if w4 = w5 . Moreover, F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (w3 −w4 )(3w1−3w2 −w5 +w6 ) ≥ 0 with equality if and only if w3 = w4 . Therefore (z1 , · · · , z6 ) = (w1 , w4, w6 , w5 , w3 , w2 ). Case 4: w1 ≤ w2 +w4 +w5 and w1 ≤ w2 +w3 −w5 . By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 . Then (z1 , · · · , z6 ) = (w1 , w4, w6 , w5 , w3 , w2 ), or (w1 , w3 , w6 , w5 , w4 , w2 ). But by Lemma 3.3, we have F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (w3 −w4 )(3w1−3w2 −w5 +w6 ) ≥ 0 with equality if and only if w3 = w4 . Therefore (z1 , · · · , z6 ) = (w1 , w4, w6 , w5 , w3 , w2 ). (6). w1 = w2 + w3 − w4 > w2 + 31 (w5 − w6 ). By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 ; or z1 ≥ z2 ≥ z3 ≥ z4 and z4 ≤ z5 ≤ z6 . Then (z1 , · · · , z6 ) = (w1 , w4 , w6 , w5 , w3 , w2 ); or (w1 , w3, w6 , w5 , w4 , w2 ); or(w1 , w5 , w6 , w4, w3 , w2 ); or (w1 , w4 , w5 , w6 , w3 , w2). But F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (w3 −w4 )(3w1−3w2 −w5 +w6 ) ≥ 0. F (w1 , w5 , w6 , w4, w3 , w2 ) − F (w1 , w4 , w6 , w5 , w3 , w2 ) = 2(w1 − w2 − w3 )(w4 − w5 ) ≤ 0. F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w4 , w5, w6 , w3 , w2 ) = (w1 +w4 −w2 −w3 )(w5 −w6 ) = 0. Therefore (z1 , · · · , z6 ) = (w1 , w4, w6 , w5 , w3 , w2 ) or (w1 , w4 , w5, w6 , w3 , w2 ). (7) w1 = w2 + 31 (w5 − w6 ) > w2 + w3 − w4 . By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 . Then (z1 , · · · , z6 ) = (w1 , w4 , w6 , w5 , w3, w2 ); or (w1 , w3 , w6 , w5 , w4 , w2); or (w1 , w5 , w6 , w4 , w3 , w2 ). But by Lemma 3.3, we have F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (w3 −w4 )(3w1−3w2 −w5 +w6 ) = 0. F (w1 , w5 , w6 , w4, w3 , w2 ) − F (w1 , w4 , w6 , w5 , w3 , w2 ) = 2(w1 − w2 − w3 )(w4 − w5 ) ≤ 0. Hence (z1 , · · · , z6 ) = (w1 , w4 , w6 , w5, w3 , w2 ) or (w1 , w3 , w6 , w5 , w4, w2 ). 17

(8). w1 = w2 + w3 − w4 = w2 + 31 (w5 − w6 ). It follows from (6) and (7) that (8) holds. (9). Assume that w2 + 13 (w5 − w6 ) ≤ w1 < w2 + w3 − w4 . We consider the following two cases: Case 1: w1 > w2 + w4 + w5 . By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 ; or z1 ≥ z2 ≥ z3 ≥ z4 and z4 ≤ z5 ≤ z6 . Hence (z1 , · · · , z6 ) = (w1 , w4 , w6 , w5 , w3 , w2); (w1 , w4 , w5 , w6 , w3 , w2 ) or (w1 , w5 , w6, w4 , w3 , w2 ). But by Lemma 3.3, we have F (w1 , w5, w6 , w4 , w3 , w2 ) − F (w1, w4 , w6 , w5 , w3 , w2 ) = 2(w1 − w2 − w3 )(w4 − w5 ) ≤ 0 with equality if and only if w4 = w5 . F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w4 , w5 , w6 , w3 , w2 ) = 2(w1 +w4 −w2 −w3 )(w5 −w6 ) ≤ 0 with equality if and only if w5 = w6 . Therefore (z1 , · · · , z6 ) = (w1 , w4, w5 , w6 , w3 , w2 ). Case 2: w1 ≤ w2 + w4 + w5 . By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 ; or z1 ≥ z2 ≥ z3 ≥ z4 and z4 ≤ z5 ≤ z6 . Hence, (z1 , · · · , z6 ) = (w1 , w4 , w6 , w5 , w3 , w2); or (w1 , w3 , w6, w5 , w4 , w2 ); (w1 , w4 , w5 , w6, w3 , w2 ) or (w1 , w5 , w6 , w4 , w3 , w2 ). But by Lemma 3.3, we have F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (3w1 −3w2 −w5 +w6 )(w3 −w4 ) ≤ 0 with equality if and only if w3 = w4 ; F (w1 , w4 , w6 , w5, w3 , w2 ) − F (w1 , w5 , w6 , w4 , w3 , w2 ) = 2(−w1 + w2 + w3 )(w4 − w5 ) ≥ 0 and F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w4 , w5 , w6, w3 , w2 ) = (w1 +w4 −w2 −w3 )(w5 −w6 ) ≥ 0 with equality if and only if w5 = w6 . Therefore (z1 , · · · , z6 ) = (w1 , w4, w5 , w6 , w3 , w2 ). Assume that w1 ≤ w2 + 31 (w5 − w6 ) < w2 + w3 − w4 . By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 ; or z1 ≥ z2 ≥ z3 ≤ z4 and z4 ≤ z5 ≤ z6 . Hence, (z1 , · · · , z6 ) = (w1 , w4 , w6, w5 , w3 , w2 ); or (w1 , w3 , w6 , w5 , w4, w2 ); or (w1 , w5 , w6 , w4 , w3 , w2); or (w1 , w4 , w5 , w6 , w3 , w2 ). But by Lemma 3.3, we have F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (3w1 −3w2 −w5 +w6 )(w3 −w4 ) ≤ 0

18

with equality if and only if w3 = w4 ; F (w1 , w5 , w6 , w4, w3 , w2 ) − F (w1 , w4 , w6 , w5 , w3 , w2 ) = 2(w1 − w2 − w3 )(w4 − w5 ) ≤ 0; and F (w1 , w4 , w5 , w6, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (3w3 −3w4 −w5 +w6 )(w1 −w2 ) ≥ 0 with equality if and only if w1 = w2 . Therefore (z1 , · · · , z6 ) = (w1 , w4, w5 , w6 , w3 , w2 ). (10). Assume that w2 + w3 − w4 ≤ w1 < w2 + 31 (w5 − w6 ). By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 ; or z1 ≥ z2 ≥ z3 ≤ z4 and z4 ≤ z5 ≤ z6 . Hence, (z1 , · · · , z6 ) = (w1 , w4 , w6 , w5 , w3 , w2); or (w1 , w3, w6 , w5 , w4 , w2 ); or (w1 , w5 , w6 , w4 , w3 , w2 ); or (w1 , w4, w5 , w6 , w3 , w2 ). But by Lemma 3.3, we have F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (3w1 −3w2 −w5 +w6 )(w3 −w4 ) ≤ 0 with equality if and only if w3 = w4 ; F (w1 , w5 , w6 , w4, w3 , w2 ) − F (w1 , w4 , w6 , w5 , w3 , w2 ) = 2(w1 − w2 − w3 )(w4 − w5 ) ≤ 0; and F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w4 , w5 , w6, w3 , w2 ) = (w1 +w4 −w2 −w3 )(w5 −w6 ) ≥ 0 with equality if and only if w5 = w6 . Therefore (z1 , · · · , z6 ) = (w1 , w3, w6 , w5 , w4 , w2 ). Assume that w1 ≤ w2 + w3 − w4 < w2 + 31 (w5 − w6 ). By (8) and (9) in Lemma 2.5, we have z1 ≥ z2 ≥ z3 and z3 ≤ z4 ≤ z5 ≤ z6 ; or z1 ≥ z2 ≥ z3 ≤ z4 and z4 ≤ z5 ≤ z6 . Hence, (z1 , · · · , z6 ) = (w1 , w4 , w6, w5 , w3 , w2 ); or (w1 , w3 , w6 , w5 , w4, w2 ); or (w1 , w5 , w6 , w4 , w3 , w2); or (w1 , w4 , w5 , w6 , w3 , w2 ). But by Lemma 3.3, we have F (w1 , w4 , w6 , w5, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (3w1 −3w2 −w5 +w6 )(w3 −w4 ) ≤ 0 with equality if and only if w3 = w4 ; F (w1 , w5 , w6 , w4, w3 , w2 ) − F (w1 , w4 , w6 , w5 , w3 , w2 ) = 2(w1 − w2 − w3 )(w4 − w5 ) ≤ 0; and F (w1 , w4 , w5 , w6, w3 , w2 )−F (w1 , w3 , w6 , w5 , w4 , w2 ) = (3w3 −3w4 −w5 +w6 )(w1 −w2 ) ≤ 0 with equality if and only if w1 = w2 . Therefore (z1 , · · · , z6 ) = (w1 , w3, w6 , w5 , w4 , w2 ). (11). w1 < w2 + w3 − w4 = w2 + 31 (w5 − w6 ). It follows from (9) and (10) that (11) holds. Proof. of Theorem 3.4. It follows from Theorem 2.3 and Lemma 3.5 that the assertion holds. 19

Suggest Documents